Let V and W be vector spaces and T: V –> W be linear. Then N(T) and R(T) are subspaces of V and W, respectively

Theorem 2.1

Let V and W be vector spaces, and let T: V –> W be linear. If B = {v1, … , vn} is a basis for V, then R(T) = span(T(B)) = span({T(v1), … , T(vn)})

Theorem 2.2

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Let V and W be vector spaces, and let T: V –> W be linear. If V is finite-dimensional, then nullity(T) + rank(T) = dim(V)

Theorem 2.3–The Dimension Theorem

Let V and W be vector spaces, and let T: V –> W be linear. Then T is one-to-one if and only if N(T) = {0}

Theorem 2.4

Let V and W be vector spaces of equal, finite dimension, and let T: V –> W be linear. Then the following are equivalent:

a. T is one-to-one

b. T is onto

c. rank(T) = dim(V)

Theorem 2.5

Let V and W be vector spaces over F, and suppose that {v1, … , vn} is a basis for V. For w1, … , wn in W, there exists exactly one linear transformation T: V –> W such that T(vi) = wi for i = 1, … , n

Theorem 2.6

Let V and W be vector spaces, and suppose that V has a finite basis {v1, … , vn}. If U, T: V –> W are linear and U(vi) = T(vi) for i = 1, … , n, then U = T

Theorem 2.6, Corollary 1

Let V and W be vector spaces over a field F, and let T, U: V –> W be linear.

a. For all a in F, aT + U is linear

b. The collection of all linear transformations from V to W is a vector space over F

Theorem 2.7

Let V and W be finite-dimensional vector spaces with ordered bases B and Y, respectively, and let T, U: V –> W be linear transformations. Then,

a. (T + U)B,Y = (T)B,Y + (U)B,Y

b. (aT)B,Y = a(T)B,Y for all scalars a

Theorem 2.8

Let V, W, and Z be vector spaces over the same field F, and let T: V –> W and U: W –> Z be linear. Then UT: V

Theorem 2.9

Let V be a vector space. Let T, U1, U2 be in L(V). Then,

a. T(U1 + U2) = T(U1) + T(U2)

b. T(U1U2) = (TU1)(U2)

c. TI = IT = T

d. a(U1U2) = (aU1)U2 = U1(aU2) for all scalars a

Theorem 2.10

Let V, W, and Z be finite-dimensional vector spaces with ordered bases A, B, and Y. Let T: V –> W and U: W –> Z be linear transformations. Then [UT]A,Y = [U]B,Y[T]A,B

Theorem 2.11

Let V be a finite-dimensional vector space with an ordered basis B. Let T, U be in L(V). Then [UT]B = [U]B[T]B

Theorem 2.11, Corollary 1

Let A be an mxn matrix, B and C be nxp matrices, and D and E by qxm matrices. Then,

a. A(B + C) = AB + AC and (D + E)A = DA + EA

b. a(AB) = (aA)B = A(aB) for any scalar a

c. ImA = A = AIn

d. If V is an n-dimensional vector space with an ordered basis B, then [Iv]B = In

Theorem 2.12

Let A be an mxn matrix, B1, B2, … , Bk be nxp matrices, C1, C2, … , Ck be qxm matrices, and a1, … , ak be scalars. Then,

A[(sum1,k)aiBi] = (sum1,k)aiABi and

[(sum1,k)aiCi]A = (sum1,k)aiCiA

Theorem 2.12, Corollary 1

Let A be an mxn matrix and B be an nxp matrix. For each j (1 < j < p) let uj and vj denote the jth columns of AB and B, respectively. Then
a. uj = Avj
b. vj = Bej, where ej is the jth standard vector of Fp

Theorem 2.13

Let V and W be finite-dimensional vector spaces having ordered bases B and Y, respectively, and let T: V –> W be linear. Then for each u in V, we have

[T(u)]Y = [T]B,Y[u]B

Theorem 2.14

Let A be an mxn matrix with entries from F. Then the left-multiplication transformation LA: Fn –> Fm is linear. Furthermore, if B is any other mxn matrix (with entries from F) and B and Y are the standard ordered bases for Fn and Fm, respectively, then we have the following properties.

a. [LA]B,Y = A

b. LA = LB if and only if A = B

c. L(A + B) = LA + LB and LaA = aLA for all a

d. If T: Fn –> Fm is linear, then there exists a unique mxn matrix C such that T = LC. In fact, C = [T]B,Y

e. If E is an nxp matrix, then LAE = LALE

f. If m = n, then LIn = LFn

Theorem 2.15

Let A, B, and C be matrices such that A(BC) is defined. Then (AB)C is also defined, and A(BC) = (AB)C; that is, matrix multiplication is associative.

Theorem 2.16

Let V and W be vector spaces, and let T: V –> W be linear and invertible. Then T^-1: W –> V is linear

Theorem 2.17

Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, dim(V) = dim(W)

Lemma, Theorem 2.18

Let V and W be finite-dimensional vector spaces with ordered bases B and Y, respectively. Let T: V –> W be linear. Then T is invertible if and only if [T]B,Y is invertible. Furthermore, [T^-1]Y,B = ([T]B,Y)^-1

Theorem 2.18

Let V be a finite-dimensional vector space with an ordered basis B, and let T: V –> V be linear. Then T is invertible if and only if [T]B is invertible. Furthermore, [T^-1]B = ([T]B)^-1

Theorem 2.18, Corollary 1

Let A be an nxn matrix. Then A is invertible if and only if LA is invertible. Furthermore, (LA)^-1 = LA^-1

Theorem 2.18, Corollary 2

Let V and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if dim(V) = dim(W)

Theorem 2.19

Let V be a vector space over F. Then V is isomorphic to Fn if and only if dim(V) = n

Theorem 2.19, Corollary 1

Let V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let B and Y be ordered bases for V and W, respectively. Then the function G: L(V,W) –> Mnxn(F), defined by G(T) = [T]B,Y for T in L(V,W), is an isomorphism

Theorem 2.20

Let V and W be finite-dimensional vector spaces of dimensions n and m, respectively. Then L(V,W) is finite-dimensional of dimension mn

Theorem 2.20, Corollary 1

For any finite-dimensional vector space V with ordered basis B, [Phi]B is an isomorphism

Theorem 2.21

Let B and B’ be two ordered bases for a finite-dimensional vector space V, and let Q = [Iv]B’,B. Then

a. Q is invertible

b. For any v in V, [v]B = Q[v]B’

Theorem 2.22

Let T be a linear operator on a finite-dimensional vector space V, and let B and B’ be ordered bases for V. Suppose that Q is the change of coordinate matrix that changes B’ coordinates into B coordinates. Then

[T]B’ = Q^-1[T]BQ

Theorem 2.23

Let A be in Mnxn(F), and let Y be an ordered basis for Fn. Then [LA]Y = Q^-1AQ, where Q is the nxn matrix whose jth column is the jth vector of Y

Theorem 2.23, Corollary 1