MATH2210 Notebook 3 Spring 2018
|
|
- Clement Mosley
- 5 years ago
- Views:
Transcription
1 MATH2210 Notebook 3 Spring 2018 prepared by Professor Jenny Baglivo c Copyright by Jenny A. Baglivo. All Rights Reserved. 3 MATH2210 Notebook Vector Spaces and Subspaces Vector Spaces Over the Reals Vector Subspaces and Spans of Finite Sets Subspaces Related to Matrices and Linear Transformations Null Space and Column Space of a Matrix Kernel and Range of a Linear Transformation Linear Independence and Bases Linearly Independence and Linear Dependence Basis of a Vector Space Bases for Null and Column Spaces Unique Representations and Coordinate Systems Unique Representation Theorem Change of Coordinates in k-space Coordinate Mapping Theorem Dimension and Rank Dimension of a Vector Space Dimension of a Vector Subspace Dimensions of Null and Column Spaces of a Matrix Rank and Nullity of a Matrix and its Transpose
2 2
3 3 MATH2210 Notebook 3 This notebook is concerned with vector space concepts in linear algebra. The notes correspond to material in Chapter 4 of the Lay textbook. 3.1 Vector Spaces and Subspaces Recall that if {v, w} R n is a linearly independent set, then the span of these vectors, Span{v, w} = {cv + dw c, d R} R n, corresponds to a plane through the origin in R n. The span of a linearly independent set {v, w} of 2 vectors is, in many ways, just like R 2 since points in the span are uniquely determined using the two coordinates (c, d). For example, since {v, w} = 1 2 1, R3 is a linearly independent set in 3-space, we know that its span is a plane through the origin in 3-space, as illustrated to the right. The grid shown on the plane corresponds to points where either c or d is an integer. This chapter explores objects that have algebraic properties just like R k for some k, and generalizes these objects. Question: Do you remember how to show that {v, w} is a linearly independent set? 3
4 3.1.1 Vector Spaces Over the Reals A vector space V is a nonempty set of objects (called vectors) on which are defined the operations of addition and scalar multiplication satisfying the following rules (axioms): 1. Closed Under Vector Addition: For each x, y V, x + y V. 2. Closed Under Scalar Multiplication: For each x V and c R, cx V. 3. Properties of Addition: Vector addition is commutative and associative. 4. Zero Vector: There is a zero vector, O V, that satisfies the following property: x + O = O + x = x for all x V. 5. Additive Inverse: For each x V there is a y V satisfying x + y = y + x = O. 6. Properties of Scalar Multiplication: Scalar multiplication satisfies c(x + y) = (cx) + (cy), (c + d)x = (cx) + (dx), (cd)x = c(dx), 1x = x, for all c, d R, x, y V. Example: R n. For each n, R n with the usual definitions of vector sum and scalar product is a vector space. In fact, R n is our model vector space. Example: M m n. Let M m n be the collection of all m n matrices with the usual definitions of matrix sum and scalar product. Then M m n satisfies the vector space axioms, where the zero vector of the matrix space is the zero matrix O of size m n. A particular example we will use often in illustrations is the vector space of 2 2 matrices: {[ ] a b M 2 2 = a, b, c, d R} c d with the usual definitions of matrix sum and scalar product. In many ways, the vector space M 2 2 behaves like the vector space R 4 with respect to the operations of addition and scalar multiplication, where a R 4 = b c a, b, c, d R d with the usual definitions of vector sum and scalar product. More generally, the vector spaces M m n and R mn have a lot in common. 4
5 Example: P n. Let P n be the collection of polynomials of degree at most n. A polynomial p P n when it takes the following form: p(t) = a 0 + a 1 t + a 2 t a n t n for all t R, for fixed constants a 0, a 1,..., a n R. Operations on P n are defined as follows: 1. Addition: Given p 1, p 2 P n, then p 1 + p 2 is the polynomial whose value at each t is the sum of the values p 1 (t) and p 2 (t): (p 1 + p 2 )(t) = p 1 (t) + p 2 (t) for each t R. (You add polynomials by adding their values at each t.) 2. Scalar Multiplication: Givenp P n and k R, then kp is the polynomial whose value at each t is k times the value p(t): (kp)(t) = k (p(t)) for each t R. (You scale a polynomial by scaling its value at each t.) The zero vector of P n is the constant function all of whose values are zero: O(t) = 0 for all t R. In many ways, the vector space P n behaves like the vector space R n+1 with respect to the operations of addition and scalar multiplication, where a 0 R n+1 a 1 =. a 0, a 1,..., a n R a n with the usual definitions of vector sum and scalar product. Note that P n consists of polynomials whose degrees are at most n, not exactly n. Thus, P 0 P 1 P n 1 P n. Thus, the vector space of constant functions, P 0 = {p(t) = a a R}, is contained within the vector space of constant and linear functions, P 1 = {p(t) = a + bt a, b R}, is contained within the vector space of constant, linear and quadratic functions, P 2 = {p(t) = a + bt + ct 2 a, b, c R}, and so forth. 5
6 Example: C[a, b]. Let [a, b] R be an interval, and let C[a, b] be the collection of continuous real-valued functions with domain [a, b]. A convenient way to write C[a, b] is as follows: Operations on C[a, b] are defined as follows: C[a, b] = {f : [a, b] R f is continuous}. 1. Addition: Given f 1, f 2 C[a, b], then f 1 + f 2 is the function whose value at each x [a, b] is the sum of the values f 1 (x) and f 2 (x): (f 1 + f 2 )(x) = f 1 (x) + f 2 (x) for each x [a, b]. (You add functions by adding their values at each x.) 2. Scalar Multiplication: Givenf C[a, b] and k R, then kf is the function whose value at each x is k times the value f(x): (kf)(x) = k (f(x)) for each x [a, b]. (You scale functions by scaling their values at each x.) The zero vector of C[a, b] is the constant function all of whose values are zero: O(x) = 0 for all x [a, b]. The vector space of continuous functions on [a, b] does not behave like the vector space R n with respect to the operations of addition and scalar multiplication for any n. Note that the vector space of continuous functions on [a, b] contains many interesting subsets, many of which are themselves vector spaces. Examples include the following: 1. Polynomials with Restricted Domains: Let P n [a, b] be the collection of polynomial functions of degree at most n with domains restricted to the interval [a, b]. Since polynomial functions are continuous, we know that P 0 [a, b] P 1 [a, b] P n [a, b] C[a, b]. Each P n [a, b] satisfies the vector space axioms. 2. Differentiable Functions on [a, b]: Let D[a, b] be the collection of differentiable real-valued functions with domain [a, b]. Since differentiable functions are continuous, we know that D[a, b] satisfies the vector space axioms. D[a, b] C[a, b]. 6
7 3.1.2 Vector Subspaces and Spans of Finite Sets Let H be a subset of the vector space V. 1. Subspace: H is said to be a subspace of V if the following three conditions are satisfied: (1) Contains Zero Vector: O H. (2) Closed under Addition: If x, y H, then x + y H. (3) Closed Under Scalar Multiplication: If x H and c R, then cx H. 2. Span of a Finite Set of Vectors: Suppose that H is the collection of all linear combinations of a finite set of vectors in V, H = {c 1 v 1 + c 2 v c k v k c i R for all i} V. Then H = Span {v 1, v 2,..., v k } satisfies the three axioms for subspaces listed above. Subspaces are vector spaces in their own right. Specifically, Theorem (Subspace). If H V satisfies the three conditions for subspaces, then H is a vector space in its own right. In particular, if H = Span {v 1, v 2,..., v k } is the span of a finite set of vectors, then H is a vector space in its own right. Note: Importance of This Theorem If H satisfies the conditions for subspaces, then H inherits all the remaining properties needed to satisfy the axioms of a full vector space. Thus, the theorem can be used to reduce the amount of work needed to determine if a collection of vectors is a vector space. A reasonable strategy to follow to determine if H is a subspace is the following: 1. Determine if H can be written as the span of a finite set of vectors. If yes, then H is a subspace and a vector space in its own right. 2. If H cannot be written as the span of a finite set of vectors, then determine if H satisfies the three axioms for spaces. If yes, then H is a subspace and a vector space in its own right. 3. Otherwise, H is neither a subspace nor a vector space in its own right. Further, to demonstrate that a subset is not a subspace, you just need to demonstrate that one of the three conditions fails. 7
8 Problem 1. Which, if any, of the following subsets are subspaces? Why? {[ ] x (a) L = y = mx} R y 2, where m is a fixed constant. {[ ] x (b) L = y = mx + b} R y 2, where m and b 0 are fixed constants. {[ ] x (c) D = x y 2 + y 2 1} R 2. 8
9 {[ ] a 0 (d) D = a, d R} M 0 d 2 2. {[ ] a b (e) H = d = a + b + c} M c d 2 2. {[ ] a b (f) S = ad bc = 0} M c d
10 (g) H = {p(t) = a + t a R} P 3. (h) Z = { p(t) = a + bt + ct 2 + dt 3 p (t) = 0 for all t } P 3. (i) E = {f : [a, b] R f is continuous and f(a) = f(b)} C[a, b]. 10
11 3.2 Subspaces Related to Matrices and Linear Transformations Null Space and Column Space of a Matrix Let A be an m n matrix. 1. Null Space of a Matrix: The null space of A is the set of solutions to the matrix equation Ax = O: Null(A) = {x Ax = O} R n. 2. Column Space of a Matrix: The column space of A is the set of all b for which Ax = b is consistent: For example, let A = [ ] a 1 a 2 a 3 = Since [ A O ] = Col(A) = {b Ax = b is consistent} R m , the null and column spaces can be written as spans as follows (please complete): 11
12 [ Problem 1. Let A = vectors. Simplify your answers as much as possible. ]. Write Null(A) and Col(A) as spans of finite sets of 12
13 3.2.2 Kernel and Range of a Linear Transformation Let T : V W be a function between vector spaces V and W. 1. Linear Transformation: T is said to be a linear transformation if the following two conditions are satisfied: (1) T Respects Addition: T (x + y) = T (x) + T (y) for every x, y V. (2) T Respects Scalar Multiplication: T (cx) = ct (x) for every x V and c R. 2. Kernel of a Linear Transformation: If T is a linear transformation, then the kernel of T is the set of all vectors in the domain V that map to the zero vector in the co-domain W : 3. Range of a Linear Transformation: Kernel(T ) = {x T (x) = O} V. If T is a linear transformation, then the range of T is the set of all vectors in the co-domain W that can be written as the image of some vector in the domain V : Range(T ) = {b b = T (x) for some x} W. Note 1: If T : V W is a linear transformation, then T (O) = O To see this (please complete), Note 2: Linear transformations when V = R n and W = R m If T : R n R m is the linear transformation with rule T (x) = Ax, then 1. The kernel of T is the same as the null space of A: Kernel(T ) = Null(A). 2. The range of T is the same as the column space of A: Range(T ) = Col(A). 13
14 Problem 2. Which, if any, of the following are linear transformations? Why? ([ ]) a b (a) T : M 2 2 R with rule T = ad bc. c d (b) T : M 2 2 M 2 2 with rule T (A) = MA, where M is a fixed 2 2 matrix. 14
15 (c) T : P 2 P 2, where the image of the polynomial p satisfying p(t) = a + bt + ct 2 for each t R, is the polynomial T (p) satisfying T (p)(t) = (a + 1) + (b + 1)t + (c + 1)t 2, for each t R. (d) T : M 2 2 M 2 2 with rule T (A) = A + A T. 15
16 Problem 3. The following three functions are linear transformations. In each case, give explicit descriptions of the kernel and range of the transformation. (a) T : P 3 P 3, where the image of p is its derivative, T (p) = p. ([ [ ] (b) T : R 2 a a + b 2b M 2 2 is the function with rule T =. b]) b a 16
17 (c) T : P 3 R 2 is the function with rule T (p) = [ ] p (0) p. (0) 3.3 Linear Independence and Bases Linearly Independence and Linear Dependence Let V be a vector space. The subset {v 1, v 2,..., v k } V is said to be linearly independent when the linear combination c 1 v 1 + c 2 v c k v k equals the zero vector only when all scalars c i are zero. That is, when c 1 v 1 + c 2 v c k v k = O c 1 = c 2 = = c k = 0. Otherwise, the set {v 1, v 2,..., v k } is said to be linearly dependent. Note that: 1. If v i = O for some i, then the set {v 1, v 2,..., v k } is linearly dependent. 2. If each v i O, then the set {v 1, v 2,..., v k } is linearly dependent if and only if one vector can be written as a linear combination of the others. 17
18 Problem 1. In each case, determine if the set is linearly independent or linearly dependent. If linearly dependent, find a dependence relationship among the v i s. (a) {v 1, v 2, v 3 } = , , R 4. 18
19 (b) {p 1, p 2, p 3 } P 2, where the functions satisfy p 1 (t) = 1 t, p 2 (t) = 1 t 2, and p 3 (t) = 1 + 2t 3t 2 for all t R. 19
20 (c) {M 1, M 2, M 3 } = {[ ] 2 1, 2 1 [ ] 3 0, 0 2 [ ]} 1 1 M Basis of a Vector Space A basis for the vector space V is a linearly independent set that spans V. Specifically, the set {v 1, v 2,..., v k } is said to be a basis for V if 1. {v 1, v 2,..., v k } is linearly independent and 2. V = Span{v 1, v 2,..., v k }. Problem 1, continued. Using the work above, (a) V = Span{v 1, v 2, v 3 } R 4 has basis. (b) V = Span{p 1, p 2, p 3 } P 2 has basis. (c) V = Span{M 1, M 2, M 3 } M 2 2 has basis. 20
21 Problem 2. In each case, find a basis for V. {[ ] 0 2 (a) V = Span, 1 0 [ ] 0 1, 0 0 [ ]} 0 2 M (b) V = {A : Each column of A has sum 0} M
22 3.3.3 Bases for Null and Column Spaces Let A be an m n matrix. Then 1. The method we use to write Null(A) as the span of a finite set of vectors automatically produces a linearly independent set, hence a basis for N ull(a). 2. The pivot columns of A form a basis for Col(A). Problem 3. Let A = [ A O ] = Use the fact that to find bases for Null(A) and Col(A). 22
23 3.4 Unique Representations and Coordinate Systems Unique Representation Theorem Bases can be used to establish coordinate systems. To begin, we consider the following theorem: Unique Representation Theorem. Let B = {b 1, b 2,..., b k } be a basis for the vector space V. Then for each v V, v = c 1 b 1 + c 2 b c k b k for unique constants c 1, c 2,..., c k. The c i s are called the coordinates of v in basis B. Proof: The proof of the unique representation theorem is straightforward. Suppose that v = c 1 b 1 + c 2 b c k b k = d 1 b 1 + d 2 b d k b k for scalars c i, d i R. Then, by subtracting the second representation from the first, we get O = (c 1 d 1 )b 1 + (c 2 d 2 )b (c k d k )b k. Now, since B is a basis, we know that c i d i = 0 (equivalently, c i = d i ) for each i. Thus, the representation is unique. Coordinate Vectors. Let v = c 1 b 1 + c 2 b c k b k be the unique representation of v in the basis B. Then c 1 c 2 [v] B =. c k Rk is the coordinate vector of v in the basis B. {[ ] [ ]} 1 0 For example, let E = {e 1, e 2 } =, be the standard basis of R [ x Further, let v = be an arbitrary vector in R y] 2. Since v = xe 1 + ye 2, the coordinate vector of v in the standard basis is the vector itself: [v] E = v. 23
24 Problem 1. Consider the basis {[ ] [ 1 1 B = {b 1, b 2 } =, 1 1 [ of R 2 x, and let v = be a vector. y] ]} Find the coordinate vector of v in basis B, [v] B. [The plot shows a grid of linear combinations of the form cb 1 + db 2, where either c or d is an integer.] Problem 2. B = {b 1, b 2 } = basis for a subspace V of R 3. 5 Let v = 0 V. Find [v] B. 4, is a [V corresponds to the xz-plane in 3-space. The plot shows a grid of linear combinations of the form cb 1 + db 2, where either c or d is an integer.] 24
25 Problem 3. B = {1 t 2, 1 + t 2 } is a basis for a subspace V of P 2 = {a + bt + ct 2 : a, b, c R}. Let p(t) = 2 + 7t 2. Find [p] B Change of Coordinates in k-space Let B = {b 1, b 2,..., b k } be a basis for R k and P B be the matrix whose columns are the b i s: P B = [ b 1 b 2 b k ]. Since the columns of P B are linearly independent, we know that P B is invertible and P B c = v = c = P 1 B v or [v] B = P 1 B v. For this reason, P B is called the change-of-coordinates matrix. {[ ] [ 1 1 Problem 1, continued. As above, let B = {b 1, b 2 } =, 1 1 ]} and v = [ x y]. Find P B, and verify that [v] B = P 1 v is the same as the vector obtained earlier. B 25
26 3.4.3 Coordinate Mapping Theorem The process of finding coordinate vectors can be viewed as a linear transformation. Specifically, Coordinate Mapping Theorem. vector space V. The function Let B = {b 1, b 2,..., b k } be a basis for the T : V R k with rule T (v) = [v] B is a linear transformation that is both one-to-one and onto. The linear transformation T is known as the coordinate mapping function. For example, B = {1, t, t 2 } is the standard basis for P 2 = {a + bt + ct 2 : a, b, c R}. With respect to this basis, the coordinate mapping function is the transformation a T : P 2 R 3 which maps p(t) = a + bt + ct 2 to T (p) = b. c Isomorphisms; Isomorphic vector spaces. An isomorphism is an invertible linear transformation T : V W between vector spaces V and W. If an isomorphism exists, then we say that the vector spaces V and W are isomorphic (have the same structure). The coordinate mapping theorem tells us that V and R k are isomorphic vector spaces. The practical implication of this theorem is that a problem involving vectors in V can be translated to R k (via the coordinate mapping function), solved in R k and translated back. Continuing with the example above, suppose that we are interested in determining if the set {p 1 (t), p 2 (t), p 3 (t)} = {1 + 2t 2, 4 + t + 5t 2, 3 + 2t} P 2 is linearly independent. We could equivalently determine if {v 1, v 2, v 3 } = 0, 1, 2 is a linearly independent set in R Since [ v 1 v 2 v 3 O ] = we know that the set of vectors in R 3 and the set of pre-images in P 2 are linearly dependent. Further, we can write p 3 (t) as a linear combination of the first two polynomials as follows:, p 3 (t) = for all t R. 26
27 Problem 4. B = {[ ] 1 0, 0 0 [ ] 0 1, 0 0 [ ] 0 0, 1 0 [ ]} 0 0 is the standard basis for M With respect to this basis, the coordinate mapping function is the transformation ( [a ] ) a T : M 2 2 R 4 b with rule T = b c d c. d Using the standard basis and coordinate mapping function, (a) Determine if {M 1, M 2, M 3 } = {[ ] [ 1 1, 3 3 ] [ 1 3, 1 5 is a linearly independent set in M 2 2. If the set is linearly dependent, express one of the matrices as a linear combination of the others. ]} 27
28 {[ 1 1 (b) Demonstrate that {A 1, A 2, A 3, A 4 } = 0 1 ] [ 1 0, 1 1 ] [ 0 1, 1 1 ] [ 1 1, 1 0 is a linearly independent set in M 2 2. [ ] 3 2 (c) Express as a linear combination of the 4 matrices given in part (b). 1 0 ]} 28
29 3.5 Dimension and Rank Dimension of a Vector Space The following theorem tells us that if a vector space V has a basis with k elements, then all bases of V must have exactly k elements. Theorem (Finite Bases). If B = {b 1, b 2,..., b k } is a basis for V, then 1. Any subset of V with more than k vectors must be linearly dependent. 2. Every basis of V must contain exactly k vectors. 3. Every linearly independent set with exactly k vectors is a basis of V. Note that each part of this theorem can be proven using the coordinate mapping theorem. k-dimensional vector spaces: The vector space V is said to be k-dimensional if V has a basis with k elements. In this case, we write dim(v ) = k ( the dimension of V equals k ). For example, given positive integers m and n, dim(r n ) =, dim(p n ) = and dim(m m n ) =. Trivial vector space: By convention, the dimension of the trivial vector space is zero, dim({o}) = 0. Note that the trivial vector space does not have a basis since {O} is linearly dependent. Infinite-dimensional vector spaces: If V is not trivial and is not spanned by a finite set of vectors, then V is said to be infinite-dimensional. Examples of infinite-dimensional vector spaces include 1. The vector space of polynomials of all orders, P = {p(t) p(t) is a polynomial function of any degree}, with the usual definitions of addition and scalar multiplication. 2. The vector space of continuous real-valued functions on the interval [a, b], C[a, b] = {f : [a, b] R f is continuous}, with the usual definitions of addition and scalar multiplication. 29
30 3.5.2 Dimension of a Vector Subspace The dimension of a subspace of V can never be more than the dimension of V. Specifically, Theorem (Subspaces). If H V is a subspace of a vector space with finite dimension k, then H has finite dimension and dim(h) dim(v ) = k. Problem 1. In each case, find the dimension of the subspace H. a + b + 2c (a) H = 2a + 2b + 4c + d b + c + d 3a + 3c + d : a, b, c, d R R 4. 30
31 (b) H = Span{p 1, p 2, p 3 } P 2, where p 1 (t) = t + t 2, p 2 (t) = 1 + t, p 3 (t) = 1 + 4t + 3t 2, for each t R Dimensions of Null and Column Spaces of a Matrix Let A be an m n matrix. The dimension of the null space corresponds to the number of free variables when solving Ax = O. The dimension of the column space corresponds to the number of pivot columns of A. Problem 2. In each case, find the dimensions of the null and column spaces of A. (a) A =
32 (b) A = Rank and Nullity of a Matrix and its Transpose Let A be an m n matrix. Then 1. Rank of a Matrix: The rank of A is the dimension of its column space: rank(a) = dim(col(a)). 2. Nullity of a Matrix: The nullity of A is the dimension of its null space: nullity(a) = dim(null(a)). Similarly, if A T is the n m transpose of the matrix A, then we let rank(a T ) = dim(col(a T )) and nullity(a T ) = dim(null(a T )). 32
33 The following theorem gives interesting relationships among these numbers: Rank Theorem. Let A be an m n matrix, and let A T be n m the transpose of the matrix A. Then 1. rank(a) + nullity(a) = n. 2. rank(a T ) + nullity(a T ) = m. 3. rank(a T ) = rank(a). Problem 3. Explain why the first part of the theorem is true. Problem 4. Let A = [ Find rank(a), rank(a T ), nullity(a), nullity(a T ). ]. 33
34 Problem 5. Let I n be the n n identity matrix and O m n be the m n zero matrix. Fill-in the following table: ] A = [I 3 O 3 2 Size of A: rank(a) = m n rank(a T ) nullity(a) nullity(a T ) [ ] I 3 I 3 A = O 2 3 O 2 3 A = O 3 2 Problem 6. Let A be an m n matrix. In each case, determine if the statment is TRUE or FALSE, and give a reason for each choice. ([ ]) A (a) Col(A) = Col. A ([ ]) A (b) Null(A) = Null. A (c) Col ( A T ) ( [A ]T ) = Col. A (d) Null ( A T ) ( [A ]T ) = Null. A 34
Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationMATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:
More informationVector space and subspace
Vector space and subspace Math 112, week 8 Goals: Vector space, subspace, span. Null space, column space. Linearly independent, bases. Suggested Textbook Readings: Sections 4.1, 4.2, 4.3 Week 8: Vector
More informationMAT 242 CHAPTER 4: SUBSPACES OF R n
MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) March 5, 2015
Midterm 1 Advanced Linear Algebra Math 4377 / 638 (Spring 215) March 5, 215 2 points 1. Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain
More informationVector space and subspace
Vector space and subspace Math 112, week 8 Goals: Vector space, subspace. Linear combination and span. Kernel and range (null space and column space). Suggested Textbook Readings: Sections 4.1, 4.2 Week
More informationMath 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence
Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the left-hand side of the coefficient matrix of a linear system A x = b which could
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationLecture 13: Row and column spaces
Spring 2018 UW-Madison Lecture 13: Row and column spaces 1 The column space of a matrix 1.1 Definition The column space of matrix A denoted as Col(A) is the space consisting of all linear combinations
More informationICS 6N Computational Linear Algebra Vector Space
ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of
More informationReview Notes for Midterm #2
Review Notes for Midterm #2 Joris Vankerschaver This version: Nov. 2, 200 Abstract This is a summary of the basic definitions and results that we discussed during class. Whenever a proof is provided, I
More informationAbstract Vector Spaces and Concrete Examples
LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture : Null and Column Spaces Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./8 Announcements Study Guide posted HWK posted Math 9Applied
More information2. (10 pts) How many vectors are in the null space of the matrix A = 0 1 1? (i). Zero. (iv). Three. (ii). One. (v).
Exam 3 MAS 3105 Applied Linear Algebra, Spring 2018 (Clearly!) Print Name: Apr 10, 2018 Read all of what follows carefully before starting! 1. This test has 7 problems and is worth 110 points. Please be
More informationMidterm #2 Solutions
Naneh Apkarian Math F Winter Midterm # Solutions Here is a solution key for the second midterm. The solutions presented here are more complete and thorough than your responses needed to be - in order to
More information1 Last time: inverses
MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a
More informationObjective: Introduction of vector spaces, subspaces, and bases. Linear Algebra: Section
Objective: Introduction of vector spaces, subspaces, and bases. Vector space Vector space Examples: R n, subsets of R n, the set of polynomials (up to degree n), the set of (continuous, differentiable)
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationSept. 26, 2013 Math 3312 sec 003 Fall 2013
Sept. 26, 2013 Math 3312 sec 003 Fall 2013 Section 4.1: Vector Spaces and Subspaces Definition A vector space is a nonempty set V of objects called vectors together with two operations called vector addition
More informationChapter 2 Subspaces of R n and Their Dimensions
Chapter 2 Subspaces of R n and Their Dimensions Vector Space R n. R n Definition.. The vector space R n is a set of all n-tuples (called vectors) x x 2 x =., where x, x 2,, x n are real numbers, together
More informationSUMMARY OF MATH 1600
SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationEXERCISE SET 5.1. = (kx + kx + k, ky + ky + k ) = (kx + kx + 1, ky + ky + 1) = ((k + )x + 1, (k + )y + 1)
EXERCISE SET 5. 6. The pair (, 2) is in the set but the pair ( )(, 2) = (, 2) is not because the first component is negative; hence Axiom 6 fails. Axiom 5 also fails. 8. Axioms, 2, 3, 6, 9, and are easily
More informationComps Study Guide for Linear Algebra
Comps Study Guide for Linear Algebra Department of Mathematics and Statistics Amherst College September, 207 This study guide was written to help you prepare for the linear algebra portion of the Comprehensive
More informationMTH 362: Advanced Engineering Mathematics
MTH 362: Advanced Engineering Mathematics Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 26, 2017 1 Linear Independence and Dependence of Vectors
More informationMath 4377/6308 Advanced Linear Algebra
2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/
More informationChapter 3. Vector spaces
Chapter 3. Vector spaces Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/22 Linear combinations Suppose that v 1,v 2,...,v n and v are vectors in R m. Definition 3.1 Linear combination We say
More informationSolutions to Math 51 First Exam April 21, 2011
Solutions to Math 5 First Exam April,. ( points) (a) Give the precise definition of a (linear) subspace V of R n. (4 points) A linear subspace V of R n is a subset V R n which satisfies V. If x, y V then
More informationMATH 235. Final ANSWERS May 5, 2015
MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationWhat is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix
Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2
More informationMath 102, Winter 2009, Homework 7
Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6
More informationDimension. Eigenvalue and eigenvector
Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,
More informationRank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Rank and Nullity MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives We have defined and studied the important vector spaces associated with matrices (row space,
More informationMath 308 Practice Test for Final Exam Winter 2015
Math 38 Practice Test for Final Exam Winter 25 No books are allowed during the exam. But you are allowed one sheet ( x 8) of handwritten notes (back and front). You may use a calculator. For TRUE/FALSE
More informationSolutions to Section 2.9 Homework Problems Problems 1 5, 7, 9, 10 15, (odd), and 38. S. F. Ellermeyer June 21, 2002
Solutions to Section 9 Homework Problems Problems 9 (odd) and 8 S F Ellermeyer June The pictured set contains the vector u but not the vector u so this set is not a subspace of The pictured set contains
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationMath 110: Worksheet 3
Math 110: Worksheet 3 September 13 Thursday Sept. 7: 2.1 1. Fix A M n n (F ) and define T : M n n (F ) M n n (F ) by T (B) = AB BA. (a) Show that T is a linear transformation. Let B, C M n n (F ) and a
More informationft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST
me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following
More informationLinear Algebra Final Exam Study Guide Solutions Fall 2012
. Let A = Given that v = 7 7 67 5 75 78 Linear Algebra Final Exam Study Guide Solutions Fall 5 explain why it is not possible to diagonalize A. is an eigenvector for A and λ = is an eigenvalue for A diagonalize
More information(i) [7 points] Compute the determinant of the following matrix using cofactor expansion.
Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det
More information2018 Fall 2210Q Section 013 Midterm Exam II Solution
08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique
More informationAbstract Vector Spaces
CHAPTER 1 Abstract Vector Spaces 1.1 Vector Spaces Let K be a field, i.e. a number system where you can add, subtract, multiply and divide. In this course we will take K to be R, C or Q. Definition 1.1.
More informationMATH 423 Linear Algebra II Lecture 12: Review for Test 1.
MATH 423 Linear Algebra II Lecture 12: Review for Test 1. Topics for Test 1 Vector spaces (F/I/S 1.1 1.7, 2.2, 2.4) Vector spaces: axioms and basic properties. Basic examples of vector spaces (coordinate
More informationShorts
Math 45 - Midterm Thursday, October 3, 4 Circle your section: Philipp Hieronymi pm 3pm Armin Straub 9am am Name: NetID: UIN: Problem. [ point] Write down the number of your discussion section (for instance,
More informationChapter 2. General Vector Spaces. 2.1 Real Vector Spaces
Chapter 2 General Vector Spaces Outline : Real vector spaces Subspaces Linear independence Basis and dimension Row Space, Column Space, and Nullspace 2 Real Vector Spaces 2 Example () Let u and v be vectors
More informationVector Spaces - Definition
Vector Spaces - Definition Definition Let V be a set of vectors equipped with two operations: vector addition and scalar multiplication. Then V is called a vector space if for all vectors u,v V, the following
More informationMath Linear algebra, Spring Semester Dan Abramovich
Math 52 0 - Linear algebra, Spring Semester 2012-2013 Dan Abramovich Fields. We learned to work with fields of numbers in school: Q = fractions of integers R = all real numbers, represented by infinite
More informationSolutions to Midterm 2 Practice Problems Written by Victoria Kala Last updated 11/10/2015
Solutions to Midterm 2 Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated //25 Answers This page contains answers only. Detailed solutions are on the following pages. 2 7. (a)
More informationReview Notes for Linear Algebra True or False Last Updated: February 22, 2010
Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationCriteria for Determining If A Subset is a Subspace
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
More information1 Systems of equations
Highlights from linear algebra David Milovich, Math 2 TA for sections -6 November, 28 Systems of equations A leading entry in a matrix is the first (leftmost) nonzero entry of a row. For example, the leading
More informationInstructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.
Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.. Recall that P 3 denotes the vector space of polynomials of degree less
More informationMath 313 Chapter 5 Review
Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row
More informationLinear Transformations: Kernel, Range, 1-1, Onto
Linear Transformations: Kernel, Range, 1-1, Onto Linear Algebra Josh Engwer TTU 09 November 2015 Josh Engwer (TTU) Linear Transformations: Kernel, Range, 1-1, Onto 09 November 2015 1 / 13 Kernel of a Linear
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this
More informationLecture 14: Orthogonality and general vector spaces. 2 Orthogonal vectors, spaces and matrices
Lecture 14: Orthogonality and general vector spaces 1 Symmetric matrices Recall the definition of transpose A T in Lecture note 9. Definition 1.1. If a square matrix S satisfies then we say S is a symmetric
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More informationDaily Update. Math 290: Elementary Linear Algebra Fall 2018
Daily Update Math 90: Elementary Linear Algebra Fall 08 Lecture 7: Tuesday, December 4 After reviewing the definitions of a linear transformation, and the kernel and range of a linear transformation, we
More information4.3 - Linear Combinations and Independence of Vectors
- Linear Combinations and Independence of Vectors De nitions, Theorems, and Examples De nition 1 A vector v in a vector space V is called a linear combination of the vectors u 1, u,,u k in V if v can be
More informationWorksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases
Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation
More informationSYMBOL EXPLANATION EXAMPLE
MATH 4310 PRELIM I REVIEW Notation These are the symbols we have used in class, leading up to Prelim I, and which I will use on the exam SYMBOL EXPLANATION EXAMPLE {a, b, c, } The is the way to write the
More informationMath 54 HW 4 solutions
Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More informationReview for Exam 2 Solutions
Review for Exam 2 Solutions Note: All vector spaces are real vector spaces. Definition 4.4 will be provided on the exam as it appears in the textbook.. Determine if the following sets V together with operations
More informationLinear Algebra (Math-324) Lecture Notes
Linear Algebra (Math-324) Lecture Notes Dr. Ali Koam and Dr. Azeem Haider September 24, 2017 c 2017,, Jazan All Rights Reserved 1 Contents 1 Real Vector Spaces 6 2 Subspaces 11 3 Linear Combination and
More informationNo books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
More information7. Dimension and Structure.
7. Dimension and Structure 7.1. Basis and Dimension Bases for Subspaces Example 2 The standard unit vectors e 1, e 2,, e n are linearly independent, for if we write (2) in component form, then we obtain
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationLinear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008
Linear Algebra Chih-Wei Yi Dept. of Computer Science National Chiao Tung University November, 008 Section De nition and Examples Section De nition and Examples Section De nition and Examples De nition
More informationMath 353, Practice Midterm 1
Math 353, Practice Midterm Name: This exam consists of 8 pages including this front page Ground Rules No calculator is allowed 2 Show your work for every problem unless otherwise stated Score 2 2 3 5 4
More informationLinear Algebra Notes. Lecture Notes, University of Toronto, Fall 2016
Linear Algebra Notes Lecture Notes, University of Toronto, Fall 2016 (Ctd ) 11 Isomorphisms 1 Linear maps Definition 11 An invertible linear map T : V W is called a linear isomorphism from V to W Etymology:
More information(a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii)
. Which of the following are Vector Spaces? (i) V = { polynomials of the form q(t) = t 3 + at 2 + bt + c : a b c are real numbers} (ii) V = {at { 2 + b : a b are real numbers} } a (iii) V = : a 0 b is
More informationChapter SSM: Linear Algebra. 5. Find all x such that A x = , so that x 1 = x 2 = 0.
Chapter Find all x such that A x : Chapter, so that x x ker(a) { } Find all x such that A x ; note that all x in R satisfy the equation, so that ker(a) R span( e, e ) 5 Find all x such that A x 5 ; x x
More informationVector Space Concepts
Vector Space Concepts ECE 174 Introduction to Linear & Nonlinear Optimization Ken Kreutz-Delgado ECE Department, UC San Diego Ken Kreutz-Delgado (UC San Diego) ECE 174 Fall 2016 1 / 25 Vector Space Theory
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationMath 217 Midterm 1. Winter Solutions. Question Points Score Total: 100
Math 7 Midterm Winter 4 Solutions Name: Section: Question Points Score 8 5 3 4 5 5 6 8 7 6 8 8 Total: Math 7 Solutions Midterm, Page of 7. Write complete, precise definitions for each of the following
More informationGeneral Vector Space (3A) Young Won Lim 11/19/12
General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
More informationTues Feb Vector spaces and subspaces. Announcements: Warm-up Exercise:
Math 2270-004 Week 7 notes We will not necessarily finish the material from a given day's notes on that day. We may also add or subtract some material as the week progresses, but these notes represent
More informationDr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science
Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated
More informationMath 54 First Midterm Exam, Prof. Srivastava September 23, 2016, 4:10pm 5:00pm, 155 Dwinelle Hall.
Math 54 First Midterm Exam, Prof Srivastava September 23, 26, 4:pm 5:pm, 55 Dwinelle Hall Name: SID: Instructions: Write all answers in the provided space This exam includes two pages of scratch paper,
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationSolutions to Math 51 Midterm 1 July 6, 2016
Solutions to Math 5 Midterm July 6, 26. (a) (6 points) Find an equation (of the form ax + by + cz = d) for the plane P in R 3 passing through the points (, 2, ), (2,, ), and (,, ). We first compute two
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More informationJim Lambers MAT 610 Summer Session Lecture 1 Notes
Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra
More informationMath 2174: Practice Midterm 1
Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider
More informationMath 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix
Math 34H EXAM I Do all of the problems below. Point values for each of the problems are adjacent to the problem number. Calculators may be used to check your answer but not to arrive at your answer. That
More informationSolutions for Math 225 Assignment #4 1
Solutions for Math 225 Assignment #4 () Let B {(3, 4), (4, 5)} and C {(, ), (0, )} be two ordered bases of R 2 (a) Find the change-of-basis matrices P C B and P B C (b) Find v] B if v] C ] (c) Find v]
More informationMidterm 1 Solutions Math Section 55 - Spring 2018 Instructor: Daren Cheng
Midterm 1 Solutions Math 20250 Section 55 - Spring 2018 Instructor: Daren Cheng #1 Do the following problems using row reduction. (a) (6 pts) Let A = 2 1 2 6 1 3 8 17 3 5 4 5 Find bases for N A and R A,
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationMath 321: Linear Algebra
Math 32: Linear Algebra T. Kapitula Department of Mathematics and Statistics University of New Mexico September 8, 24 Textbook: Linear Algebra,by J. Hefferon E-mail: kapitula@math.unm.edu Prof. Kapitula,
More information3.3 Linear Independence
Prepared by Dr. Archara Pacheenburawana (MA3 Sec 75) 84 3.3 Linear Independence In this section we look more closely at the structure of vector spaces. To begin with, we restrict ourselves to vector spaces
More information