# (II.B) Basis and dimension

Size: px
Start display at page:

Transcription

1 (II.B) Basis and dimension How would you explain that a plane has two dimensions? Well, you can go in two independent directions, and no more. To make this idea precise, we formulate the DEFINITION 1. A (finite) subset { v 1,..., v m } of a vector space V (over a field F) is a basis of V if (a) they are linearly independent (over F), and (b) they span V (over F). 1 The over F means that the linear combinations are understood to be of the form α i v i, with α i F. As usual, our default will be F = R. EXAMPLE 2. A basis of R m is given by the standard basis vectors {ê i } m. Of course, one could say the same for any field: the standard basis vectors are also basis for C m, viewed as a vector space over C. But what if we view C m as a vector space over R? Then you need the 2m vectors ê 1,..., ê m and iê 1,..., iê m! (Why?) EXAMPLE 3. Let P 2 (x, y) denote the vector space of quadratic polynomials in two variables x and y, with real coefficients. This has the monomial basis {ρ 2 i (x, y)}6 = {1, x, y, xy, x2, y 2 }. Similarly, {ρ 3 i (x, y)}10 = {1, x, y, xy, x2, y 2, x 2 y, y 2 x, x 3, y 3 } gives a basis for the space P 3 (x, y) of cubic polynomials. We ll give a geometric application of this example below. 1 I am not saying that you can t have infinite bases; indeed, non-finitely-generated vector spaces (like P (x) := all polynomials) may have such a basis. I m just not defining or discussing them at this stage. 1

2 2 (II.B) BASIS AND DIMENSION EXAMPLE 4. Let A be an m n matrix, and V be the vector subspace of R n consisting of solutions to A x = 0. How do you compute a basis of V? By making use of the row-reduction algorithm, of course! For instance, we might have A = rre f (A) = The free variables are x 2 and x 4, and an arbitrary solution is given by plugging arbitrary values in for these and solving for the pivot variables x 1, x 3, x 5. To get a basis, you plug in (x 2 = 1, x 4 = 0) and (x 2 = 0, x 4 = 1) to obtain , More generally, the i th basis element is given by setting the i th free variable to 1 and the rest to zero. EXAMPLE 5. Suppose A is an invertible m m matrix. I claim that its columns { c i } m are a basis for Rm. To establish this, check that: { c i } are linearly independent : γ 1 c γ m c m = 0 = A γ = 0 = γ = 0 (since A is invertible). { c i } span R m : show every y R m can be written y = γ 1 c γ m c m (= A γ). This is easy: just put γ = A 1 y. We say that V is f initely generated if some finite subset { v 1,..., v m } spans V. Here is the precise reason why dimension makes sense: PROPOSITION 6. If V is finitely generated then (i) it has a (finite) basis, and (ii) any 2 bases of V have the same number of elements.

3 (II.B) BASIS AND DIMENSION 3 DEFINITION 7. We define the dimension dim F (V) of V over F to be the number of elements in (ii). 2 We get (i) by going through the generating subset { v 1,..., v m } and throwing out any v i that is a linear combination of the earlier vectors. What remains is a basis. In order to show (ii) we first prove the following LEMMA 8. If v 1,..., v m span V then any independent set V has no more than m vectors. PROOF. Consider any set u 1,..., u n, n > m (of vectors in V ). Since v 1,..., v m span V, each u j is a linear combination of the v i : u j = m A ij v i, for A m n. Since m < n, there is a nontrivial solution to A x = 0, some nonzero x R n. That is, and so n x j u j = j=1 n A ij x j = 0 for each i, j=1 n m x j A ij v i = j=1 ( m n x j A ij ) v i = j=1 m 0 v i = 0. We conclude that { u j } n j=1 is not an independent set! Now let { v 1,..., v m } and { w 1,..., w n } be two bases for V. The { v i } span V and the { w j } are independent, so the Lemma = n m. The same argument with { v i } and { w j } swapped = n m, and so we ve proved the Proposition. We can rephrase the Lemma as follows: if dim(v) = m then (a) less than m vectors cannot span V, while (b) more than m vectors are linearly dependent. A Geometric Application. Given 5 points A, B, C, D, E in general position in the plane, there is a unique conic passing through 2 The F is typically omitted when there is no ambiguity. That said, in the first example we had dim C (C n ) = n while dim R (C n ) = 2n.

4 4 (II.B) BASIS AND DIMENSION them. Here s why: the equation of a conic Q has the form 0 = f Q (x, y) = 6 a i ρ 2 i (x, y) = a 1 + a 2 x + a 3 y + a 4 xy + a 5 x 2 + a 6 y 2, and the statement that it passes through A is 0 = f Q (x A, y A ) = a i ρ i (x A, y A ). Continuing this for B thru E we get 5 equations in 6 unknowns a i, a system which generically 3 has a line of solutions: if (a 1,..., a 6 ) solves the system then so does (c a 1,..., c a 6 ). But this just gives multiples of the same f Q, which won t change the shape of the curve, and so there is really only one nontrivial solution: through five general points in the plane there exists a unique conic Q. Similarly, given 8 points A, B, C, D, E, P 1, P 2, P 3 what can we say about the vector space of cubics passing through them (i.e. of cubic polynomials in (x,y) vanishing at all 8 points)? First of all, the vector space of all cubics 10 a i ρ 3 i (x, y) = a 1 + a 2 x + a 3 y a 9 x 3 + a 10 y 3 should be thought of in terms of the coefficient vector a. The 8 constraints give a linear system 0 = 10 a iρ 3 i (x A, y A ). 0 = 10 a iρ 3 i (x 8 equations P 3, y P3 ) whose space of solutions in general (i.e., under the assumption of maximal rank[=8]) is two-dimensional there are 8 leading 1 s in rre f of the 8 10 [ρ] matrix, and therefore 2 parameters to choose freely. So start with A, B, C, D, E in general position ; we d like to construct points on the (unique) conic through them, using a straightedge. Begin by drawing the lines AB, DE, CD, and BC ; label 3 more precisely, generically means here that the matrix [ρ] (with rows indexed by A thru E, columns by i = 1 thru 6 ) is of maximal rank (= 5 ).

5 (II.B) BASIS AND DIMENSION 5 AB DE by P 1. (Q is drawn in a background color because we don t know what it looks like yet.) Q B C A D E BC AB CD DE P 1 Now draw (almost) any line l through A the choice is free except that l shouldn t go through B, C, D, or E. All the rest of the construction depends on this choice; in particular the final point q depends on l, and so by varying l we can vary q and in this way (we hope) construct enough points on Q to get an idea of what it looks like. Label l CD =: P 2 ; then draw P 1 P 2 and label P 1 P 2 BC =: P 3 ; finally set q := EP 3 l. We must prove q Q. Adding dotted lines to our diagram for the lines depending on l, Q B C A q E D BC AB l CD EP 3 DE P 1 P 2 P 3

6 6 (II.B) BASIS AND DIMENSION Now consider the following three cubic equations (where e.g. f AB (x, y) = 0 is the linear equation of the line AB, and so on): f 1 (x, y) := f Q (x, y) f P1 P 2 (x, y) = 0, f 2 (x, y) := f AB (x, y) f CD (x, y) f EP3 (x, y) = 0, f 3 (x, y) := f l (x, y) f BC (x, y) f DE (x, y) = 0. The graphical solutions of these equations are (respectively) Q P 1 P 2, AB CD EP 3, l BC DE. All 3 of these unions contain the 8 points A, B, C, D, E, P 1, P 2, P 3 ; that is, the cubic polynomials f 1, f 2, and f 3 each vanish at all 8 points. Above we explained that the (vector) space of such cubic polynomials was 10 8 = 2 dimensional; it follows that 3 vectors f 1, f 2, f 3 cannot be linearly independent and there is a nontrivial relation α f 1 (x, y) + β f 2 (x, y) + γ f 3 (x, y) = 0. Now we extract the information we want (q Q ) from this relation. First of all α cannot be zero. Otherwise β f 2 = γ f 3 are multiples of each other, which means that AB CD EP 3 = l BC DE. That would imply l was either AB, CD, or EP 3 ; and we said l cannot pass through B, C, D, or E so this gives a contradiction. So we may divide by α = f 1 (x, y) = β α f 2(x, y) γ α f 3(x, y). Now since q is contained in both EP 3 and l, f 2 (x, y) and f 3 (x, y) both vanish at q = (x q, y q ). (This is because f 2 contains a factor of f EP and f 3 contains a factor of f l.) According to the relation, therefore also f 1 = 0 at q ; this means that either f Q or f P1 P 2 vanishes at q, and so q Q or P 1 P 2. Suppose the latter: if q P 1 P 2 then the three lines P 1 P 2, EP 3, l are the same, and so E l. But once again, we disallowed such a choice of l at the outset. So q Q, and we re done. Another way of thinking of this result is that intercepts of opposite edges of a hexagon inscribed in a conic lie on line. This is a theorem proved by Pascal in 1639 known as his mystic hexagon!

7 Exercises EXERCISES 7 (1) Let V be a vector space over a subfield F of the complex numbers. Suppose u, v, w are linearly independent vectors in V. Prove that ( u + v), ( v + w), and ( w + u) are linearly independent. (2) Let V be the set of all 2 2 matrices A with complex entries which satisfy A 11 + A 22 = 0. (a) Show that V is a vector space over the field of real numbers, with the usual operations of matrix addition and multiplication of a matrix by a scalar. (b) Find a basis for this vector space. (c) Let W be the set of all matrices A in V such that A 21 = A 12 (the bar denotes complex conjugation). Prove that W is a subspace of V and find a basis for W. (3) Prove that the space of all m n matrices over the field F has dimension mn, by exhibiting a basis for this space.

### (II.D) There is exactly one rref matrix row-equivalent to any A

IID There is exactly one rref matrix row-equivalent to any A Recall that A is row-equivalent to B means A = E M E B where E i are invertible elementary matrices which we think of as row-operations In ID

### Definition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition

6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition

### Chapter 1. Vectors, Matrices, and Linear Spaces

1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution

### We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true.

Dimension We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Lemma If a vector space V has a basis B containing n vectors, then any set containing more

### Abstract 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.

### Chapter 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

### 6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and

6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis

### 2018 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

### 1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.

1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is

### Math 205, Summer I, Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b. Chapter 4, [Sections 7], 8 and 9

Math 205, Summer I, 2016 Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b Chapter 4, [Sections 7], 8 and 9 4.5 Linear Dependence, Linear Independence 4.6 Bases and Dimension 4.7 Change of Basis,

### Lecture 9: Vector Algebra

Lecture 9: Vector Algebra Linear combination of vectors Geometric interpretation Interpreting as Matrix-Vector Multiplication Span of a set of vectors Vector Spaces and Subspaces Linearly Independent/Dependent

### MATH 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.

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda 4. BASES AND DIMENSION

4. BASES AND DIMENSION Definition Let u 1,..., u n be n vectors in V. The vectors u 1,..., u n are linearly independent if the only linear combination of them equal to the zero vector has only zero scalars;

### Unit 2, Section 3: Linear Combinations, Spanning, and Linear Independence Linear Combinations, Spanning, and Linear Independence

Linear Combinations Spanning and Linear Independence We have seen that there are two operations defined on a given vector space V :. vector addition of two vectors and. scalar multiplication of a vector

### 4.6 Bases and Dimension

46 Bases and Dimension 281 40 (a) Show that {1,x,x 2,x 3 } is linearly independent on every interval (b) If f k (x) = x k for k = 0, 1,,n, show that {f 0,f 1,,f n } is linearly independent on every interval

### Vector Spaces. 9.1 Opening Remarks. Week Solvable or not solvable, that s the question. View at edx. Consider the picture

Week9 Vector Spaces 9. Opening Remarks 9.. Solvable or not solvable, that s the question Consider the picture (,) (,) p(χ) = γ + γ χ + γ χ (, ) depicting three points in R and a quadratic polynomial (polynomial

### MATH 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:

### Answers in blue. If you have questions or spot an error, let me know. 1. Find all matrices that commute with A =. 4 3

Answers in blue. If you have questions or spot an error, let me know. 3 4. Find all matrices that commute with A =. 4 3 a b If we set B = and set AB = BA, we see that 3a + 4b = 3a 4c, 4a + 3b = 3b 4d,

### Chapter 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

### MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction

MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example

### The 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

### Study 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

### 1 Last time: determinants

1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)

### Chapter 1: Systems of Linear Equations

Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where

### Linear Algebra. Preliminary Lecture Notes

Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........

### ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2018 LECTURE NOTES 3 ISSUED 24 FEBRUARY 2018 1 Gaussian elimination Let A be an (m n)-matrix Consider the following row operations on A (1) Swap the positions any

### Chapter 3. More about Vector Spaces Linear Independence, Basis and Dimension. Contents. 1 Linear Combinations, Span

Chapter 3 More about Vector Spaces Linear Independence, Basis and Dimension Vincent Astier, School of Mathematical Sciences, University College Dublin 3. Contents Linear Combinations, Span Linear Independence,

### MTH 35, SPRING 2017 NIKOS APOSTOLAKIS

MTH 35, SPRING 2017 NIKOS APOSTOLAKIS 1. Linear independence Example 1. Recall the set S = {a i : i = 1,...,5} R 4 of the last two lectures, where a 1 = (1,1,3,1) a 2 = (2,1,2, 1) a 3 = (7,3,5, 5) a 4

### IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

### Linear Algebra, Summer 2011, pt. 3

Linear Algebra, Summer 011, pt. 3 September 0, 011 Contents 1 Orthogonality. 1 1.1 The length of a vector....................... 1. Orthogonal vectors......................... 3 1.3 Orthogonal Subspaces.......................

### Math 54. Selected Solutions for Week 5

Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3

### MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS

MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS 1. HW 1: Due September 4 1.1.21. Suppose v, w R n and c is a scalar. Prove that Span(v + cw, w) = Span(v, w). We must prove two things: that every element

### 4 Hilbert s Basis Theorem and Gröbner basis

4 Hilbert s Basis Theorem and Gröbner basis We define Gröbner bases of ideals in multivariate polynomial rings and see how they work in tandem with the division algorithm. We look again at the standard

### Chapter 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

### ARE211, Fall2012. Contents. 2. Linear Algebra (cont) Vector Spaces Spanning, Dimension, Basis Matrices and Rank 8

ARE211, Fall2012 LINALGEBRA2: TUE, SEP 18, 2012 PRINTED: SEPTEMBER 27, 2012 (LEC# 8) Contents 2. Linear Algebra (cont) 1 2.6. Vector Spaces 1 2.7. Spanning, Dimension, Basis 3 2.8. Matrices and Rank 8

### Linear 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:

### Vector Spaces and Dimension. Subspaces of. R n. addition and scalar mutiplication. That is, if u, v in V and alpha in R then ( u + v) Exercise: x

Vector Spaces and Dimension Subspaces of Definition: A non-empty subset V is a subspace of if V is closed under addition and scalar mutiplication. That is, if u, v in V and alpha in R then ( u + v) V and

### Math 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

### 3.2 Constructible Numbers

102 CHAPTER 3. SYMMETRIES 3.2 Constructible Numbers Armed with a straightedge, a compass and two points 0 and 1 marked on an otherwise blank number-plane, the game is to see which complex numbers you can

### Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example,

Exam 2 Solutions. Let V be the set of pairs of real numbers (x, y). Define the following operations on V : (x, y) (x, y ) = (x + x, xx + yy ) r (x, y) = (rx, y) Check if V together with and satisfy properties

### Section 1.5. Solution Sets of Linear Systems

Section 1.5 Solution Sets of Linear Systems Plan For Today Today we will learn to describe and draw the solution set of an arbitrary system of linear equations Ax = b, using spans. Ax = b Recall: the solution

### Vector Spaces. (1) Every vector space V has a zero vector 0 V

Vector Spaces 1. Vector Spaces A (real) vector space V is a set which has two operations: 1. An association of x, y V to an element x+y V. This operation is called vector addition. 2. The association of

### Linear Algebra. Min Yan

Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

### 3. Vector spaces 3.1 Linear dependence and independence 3.2 Basis and dimension. 5. Extreme points and basic feasible solutions

A. LINEAR ALGEBRA. CONVEX SETS 1. Matrices and vectors 1.1 Matrix operations 1.2 The rank of a matrix 2. Systems of linear equations 2.1 Basic solutions 3. Vector spaces 3.1 Linear dependence and independence

### Chapter 4 & 5: Vector Spaces & Linear Transformations

Chapter 4 & 5: Vector Spaces & Linear Transformations Philip Gressman University of Pennsylvania Philip Gressman Math 240 002 2014C: Chapters 4 & 5 1 / 40 Objective The purpose of Chapter 4 is to think

### Math 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.

### Vector 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

### Systems of Linear Equations

LECTURE 6 Systems of Linear Equations You may recall that in Math 303, matrices were first introduced as a means of encapsulating the essential data underlying a system of linear equations; that is to

### Chapter 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

### Review of Linear Algebra

Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

### Outline. Linear maps. 1 Vector addition is commutative: summands can be swapped: 2 addition is associative: grouping of summands is irrelevant:

Outline Wiskunde : Vector Spaces and Linear Maps B Jacobs Institute for Computing and Information Sciences Digital Security Version: spring 0 B Jacobs Version: spring 0 Wiskunde / 55 Points in plane The

### which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar.

It follows that S is linearly dependent since the equation is satisfied by which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. is

### 1.8 Dual Spaces (non-examinable)

2 Theorem 1715 is just a restatement in terms of linear morphisms of a fact that you might have come across before: every m n matrix can be row-reduced to reduced echelon form using row operations Moreover,

### Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.

Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]

### Math 110, Spring 2015: Midterm Solutions

Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make

### div(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:

Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.

### Linear Algebra. Preliminary Lecture Notes

Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date May 9, 29 2 Contents 1 Motivation for the course 5 2 Euclidean n dimensional Space 7 2.1 Definition of n Dimensional Euclidean Space...........

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 48 RAVI VAKIL CONTENTS 1. A little more about cubic plane curves 1 2. Line bundles of degree 4, and Poncelet s Porism 1 3. Fun counterexamples using elliptic curves

### DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

### OHSX XM511 Linear Algebra: Multiple Choice Exercises for Chapter 2

OHSX XM5 Linear Algebra: Multiple Choice Exercises for Chapter. In the following, a set is given together with operations of addition and scalar multiplication. Which is not a vector space under the given

### Dimension. 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,

### Linear Algebra, Summer 2011, pt. 2

Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................

### Abstract 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.

### (I.D) Solving Linear Systems via Row-Reduction

(I.D) Solving Linear Systems via Row-Reduction Turning to the promised algorithmic approach to Gaussian elimination, we say an m n matrix M is in reduced-row echelon form if: the first nonzero entry of

### Solution to Homework 1

Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### GEOMETRY OF MATRICES x 1

GEOMETRY OF MATRICES. SPACES OF VECTORS.. Definition of R n. The space R n consists of all column vectors with n components. The components are real numbers... Representation of Vectors in R n.... R. The

### Linear Algebra Massoud Malek

CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

### Linear algebra and differential equations (Math 54): Lecture 10

Linear algebra and differential equations (Math 54): Lecture 10 Vivek Shende February 24, 2016 Hello and welcome to class! As you may have observed, your usual professor isn t here today. He ll be back

### LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS

LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in

### Matrices and RRE Form

Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

### MATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic.

MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam- You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram

### 2. Intersection Multiplicities

2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.

### This last statement about dimension is only one part of a more fundamental fact.

Chapter 4 Isomorphism and Coordinates Recall that a vector space isomorphism is a linear map that is both one-to-one and onto. Such a map preserves every aspect of the vector space structure. In other

### BASIC NOTIONS. x + y = 1 3, 3x 5y + z = A + 3B,C + 2D, DC are not defined. A + C =

CHAPTER I BASIC NOTIONS (a) 8666 and 8833 (b) a =6,a =4 will work in the first case, but there are no possible such weightings to produce the second case, since Student and Student 3 have to end up with

### Linear Algebra MAT 331. Wai Yan Pong

Linear Algebra MAT 33 Wai Yan Pong August 7, 8 Contents Linear Equations 4. Systems of Linear Equations..................... 4. Gauss-Jordan Elimination...................... 6 Linear Independence. Vector

### Math 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,...,

### Lecture Notes 1: Vector spaces

Optimization-based data analysis Fall 2017 Lecture Notes 1: Vector spaces In this chapter we review certain basic concepts of linear algebra, highlighting their application to signal processing. 1 Vector

### Math 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,

### The Gauss-Jordan Elimination Algorithm

The Gauss-Jordan Elimination Algorithm Solving Systems of Real Linear Equations A. Havens Department of Mathematics University of Massachusetts, Amherst January 24, 2018 Outline 1 Definitions Echelon Forms

### LECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK)

LECTURE 6: VECTOR SPACES II (CHAPTER 3 IN THE BOOK) In this lecture, F is a fixed field. One can assume F = R or C. 1. More about the spanning set 1.1. Let S = { v 1, v n } be n vectors in V, we have defined

### MATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005

MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all

### Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2

Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch

### Review 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

### Tangent spaces, normals and extrema

Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

### Math 240, 4.3 Linear Independence; Bases A. DeCelles. 1. definitions of linear independence, linear dependence, dependence relation, basis

Math 24 4.3 Linear Independence; Bases A. DeCelles Overview Main ideas:. definitions of linear independence linear dependence dependence relation basis 2. characterization of linearly dependent set using

### 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 +...

### Mathematics Department Stanford University Math 61CM/DM Inner products

Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector

### Chapter 2: Matrix Algebra

Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry

### 1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

### Linear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall 2011

Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 1A: Vector spaces Fields

### Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

### Solution Set 7, Fall '12

Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det

### x 2 For example, Theorem If S 1, S 2 are subspaces of R n, then S 1 S 2 is a subspace of R n. Proof. Problem 3.

.. Intersections and Sums of Subspaces Until now, subspaces have been static objects that do not interact, but that is about to change. In this section, we discuss the operations of intersection and addition

### COMPLEX ALGEBRAIC SURFACES CLASS 15

COMPLEX ALGEBRAIC SURFACES CLASS 15 RAVI VAKIL CONTENTS 1. Every smooth cubic has 27 lines, and is the blow-up of P 2 at 6 points 1 1.1. An alternate approach 4 2. Castelnuovo s Theorem 5 We now know that

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### Chapter 6: Orthogonality

Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products