Definitions for Quizzes

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1 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 it need to be a part of your definition.] 1 Linear Equations Definition 1. A matrix is in reduced row-echelon form (RREF) if it satisfies all of the following conditions: a) If a row has nonzero entries, then the first nonzero entry is a 1, [called a leading 1]. b) If a column contains a leading 1, then all the other entries in that column are 0. c) If a row contains a leading 1, then each row above it contains a leading 1 further to the left. Definition 2 (1.3.2). The rank of a matrix A, written rank(a), is the number of leading 1 s in rref(a). Definition 3 (1.3.9). A vector b R n is called a linear combination of the vectors v 1,..., v m in R n if there exist scalars x 1,..., x m such that b = x 1 v x m v m. [Note that Ax is a linear combination of the columns of A. By convention, 0 is considered to be the unique linear combination of the empty set of vectors.] 2 Linear Transformations Definition 4 (2.1.1). A function T : R m R n is called a linear transformation if there exists an n m matrix A such that T (x) = Ax for all vectors x in R m. Definition 5. For a function T : X Y, X is called the domain and Y is called the target. A function T : X Y is called one-to-one if for any y Y there is at most one input x X such that T (x) = y [(different inputs give different outputs)]. A function T : X Y is called onto if for any y Y there is at least one input x X such that T (x) = y [(every target element is an output)]. A function T : X Y is called invertible if for any y Y there is exactly one x X such that T (x) = y. [Note that a function is invertible if and only if it is both one-to-one and onto.] 1

2 3 Subspaces of R n and Their Dimensions Definition 6 (3.1.1). The image of a function T : X Y is its set of outputs: im(t ) = {T (x) : x X}, [a subset of the target Y. Note that T is onto if and only if im(t ) = Y.] [For a linear transformation T : R m R n, the image is a subset of the target R n.] im(t ) = {T (x) : x R m }, Definition 7 (3.1.2). The set of all linear combinations of the vectors v 1,..., v m in R n is called their span: span(v 1, v 2,..., v m ) = {c 1 v c m v m : c 1,..., c m R}. [If span(v 1, v 2,..., v m ) = W for some subset W of R n, we say that the vectors v 1,..., v m span W. Thus span can be used as a noun or as a verb.] Definition 8 (3.1.1). The kernel of a linear transformation T : R m R n is its set of zeros: [a subset of the domain R m ]. ker(t ) = {x R m : T (x) = 0}, Definition 9 (3.2.6). A linear relation among the vectors v 1,..., v m R n is an equation of the form c 1 v c m v m = 0 for scalars c 1,..., c m R. [If c 1 = = c m = 0, the relation is called trivial, while if at least one of the c i in nonzero, the relation is nontrivial.] Definition 10 (3.2.1). A subset W of a vector space R n is called a subspace of R n if it has the following three properties: a) contains zero vector: 0 W. b) closed under addition: If w 1, w 2 W, then w 1 + w 2 W. c) closed under scalar multiplication: If w W and k R, then kw W. [Property a is needed only to assure that W is nonempty. If W contains any vector w, then it also contains 0w = 0, by property c. Properties b and c are together equivalent to W being closed under linear combinations.] Definition 11 (3.2.3). Let v 1,..., v m R n. a) A vector v i in the list v 1,..., v m is redundant if it is a linear combination of the preceding vectors v 1,..., v i 1. [Note that v 1 is redundant if and only if it equals 0, the unique linear combination of the empty set of vectors.] b) The vectors v 1,..., v m are called linearly independent (LI) if none of them are redundant. [Otherwise, they are linearly dependent (LD).] 2

3 c) The vectors v 1,..., v m form a basis of a subspace V of R n if they span V and are linearly independent. Definition 12 (3.3.3). The number of vectors in a basis of a subspace V of R n is called the dimension of V, denoted dim(v ). Definition 13. The nullity of a matrix A, written nullity(a), is the dimension of the kernel of A. Definition 14 (3.4.5). Given two n n matrices A and B, we say that A is similar to B, abbreviated A B, if there exists an invertible matrix S such that 4 Linear Spaces AS = SB or, equivalently, B = S 1 AS. Definition 15 (4.2.1). Let V and W be linear spaces. A function f : V W is called a linear transformation if, for all f, g V and k R, and [For a linear transformation T : V W, we let T (f + g) = T (f) + T (g) and T (kf) = kt (f). im(t ) = {T (f) : f V } ker(t ) = {f V : T (f) = 0}. Then im(t ) is a subspace of the target W and ker(t ) is a subspace of the domain V, so im(t ) and ker(t ) are each linear spaces.] [If the image of T is finite dimensional, then dim(im T ) is called the rank of T, and if the kernel of T is finite dimensional, then dim(ker T ) is called the nullity of T.] Definition 16 (4.2.2). An invertible linear transformation T is called an isomorphism [(from the Greek for same structure ). The linear space V is said to be isomorphic to the linear space W, written V W, if there exists an isomorphism T : V W.] 5 Orthogonality and Least Squares Definition 17 (5.1.1). Two vectors v, w R n are called perpendicular or orthogonal if v w = 0. A vector x R n is orthogonal to a subspace V R n if x is orthogonal to all vectors v V. Definition 18 (5.1.1). The length (or magnitude or norm) of a vector v R n is v = v v. A vector u R n is called a unit vector if its length is 1 [(i.e., u = 1 or u u = 1)]. 3

4 Definition 19 (5.1.2). The vectors u 1,..., u m R n are called orthonormal if they are all unit vectors and all orthogonal to each other: { 1 if i = j u i u j = 0 if i j. Definition 20 (5.1.12). [By the Cauchy-Schwarz inequality, angle between two nonzero vectors x, y R n to be [With this definition, we have the formula θ = arccos x y x y. x y = x y cos θ x y x y for the dot product in terms of the lengths of two vectors and the angle between them.] = x y x y 1], so we may define the Definition 21 (5.3.1). A linear transformation T : R n R n is called orthogonal if it preserves the length of vectors: T (x) = x, for all x R n. [If T (x) = Ax is an orthogonal transformation, we say that A is an orthogonal matrix.] Definition 22 (5.3.5). For an m n matrix A, the transpose A T of A is the n m matrix whose ijth entry is the jith entry of A: [A T ] ij = A ji. [The rows of A become the columns of A T, and the columns of A become the rows of A T.] A square matrix A is symmetric if A T = A and skew-symmetric if A T = A. Definition 23 (5.4.4). Let A be an n m matrix. Then a vector x R m is called a least-squares solution of the system Ax = b if the distance between Ax and b is as small as possible: Definition 24 (5.5.2). b Ax b Ax for all x R m. The norm (or magnitude) of an element f of an inner product space is f = f, f. Two elements f, g of an inner product space are called orthogonal (or perpendicular) if f, g = 0. The distance between two elements of an inner product space is defined to be [the norm of their difference]: dist(f, g) = f g. The angle θ between two elements f, g of an inner product space is defined by the formula ( ) f, g θ = cos 1. f g 4

5 6 Determinants Definition 25 (6.3.2). An orthogonal matrix A with det A = 1 is called a rotation matrix, [and the linear transformation T (x) = Ax is called a rotation]. Definition 26. The m-parallelepiped defined by the vectors v 1,..., v m R n is the set of all vectors in R n of the form c 1 v c m v m, where 0 c i 1. [A 2-parallelepiped is also called a parallelogram.] The m-volume V (v 1,..., v m ) of this m-parallelepiped is defined to be V (v 1,..., v m ) = v 1 v 2 v m. [In the case m = n, this is just det A, where A is the square matrix with columns v 1,..., v n R n.] 7 Eigenvalues and Eigenvectors Definition 27 (7.1.1). Let A be an n n matrix. A nonzero vector v R n is called an eigenvector of A if [Av is a scalar multiple of v, i.e.,] Av = λv for some scalar λ. [The scalar λ is called the eigenvalue of A associated with the eigenvector v. λ-eigenvector.] We sometimes call v a Definition 28 (7.2.6). An eigenvalue λ 0 of a square matrix A has algebraic multiplicity k if [λ 0 is a root of multiplicity k of the characteristic polynomial f A (λ), meaning that we can write] f A (λ) = (λ 0 λ) k g(λ) for some polynomial g(λ) with g(λ 0 ) 0. [We write AM(λ 0 ) = k.] Definition 29 (7.3.1). Let λ be an eigenvalue of an n n matrix A. The λ-eigenspace of A, denoted E λ, is defined to be E λ = ker(a λi n ) [or] = {v R n : Av = λv} [= {λ-eigenvectors of A} {0}]. Definition 30 (7.3.2). The dimension of the λ-eigenspace E λ multiplicity of λ, [written GM(λ). We have = ker(a λi n ) is called the geometric GM(λ) = dim(e λ ) = dim(ker(a λi n )) = nullity(a λi n ) = n rank(a λi n )]. 5

6 Definition 31 (7.3.3). Let A be an n n matrix. A basis of R n consisting of eigenvectors of A is called an eigenbasis for A. Definition 32 (7.4.2). Consider a linear transformation T : R n R n given by T (x) = Ax. T is called diagonalizable if there exists a basis D of R n such that the D-matrix of T is diagonal. A is called diagonalizable if A is similar to some diagonal matrix D, i.e., if there exists an invertible matrix S such that S 1 AS is diagonal. 6

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