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Chapter 2 – Linear Transformations and Matrices

Per-Olof [email protected]

Department of Mathematics

University of California, Berkeley

Math 110 Linear Algebra



Linear Transformations

Definition

We call a function T : V → W a linear transformation from V to W if, for all x, y ∈ V and c ∈ F , we have

(a) T(x + y) = T(x) + T(y) and

(b) T(cx) = cT(x)

1 If T is linear, then T(0 ) = 0

2 T is linear ⇐⇒ T(cx + y) = cT(x) + T(y) ∀x, y ∈ V, c ∈ F

3 If T is linear, then T(x − y) = T(x) − T(y) ∀x, y ∈ V

4 T is linear ⇐⇒ for x1, . . . , xn ∈ V and a1, . . . , an ∈ F ,

T (

ni=1 aixi) =

ni=1 aiT(xi)

Special linear transformations

The identity transformation IV : V → V: IV(x) = x, ∀x ∈ V

The zero transformation T0 : V → W: T0(x) = 0 ∀x ∈ V



Null Space and Range

Definition

For linear T : V → W, the null space (or kernel ) N(T) of T is theset of all x ∈ V such that T(x) = 0 : N(T) = {x ∈ V : T(x) = 0 }The range (or image ) R(T) of T is the subset of W consisting of all images of vectors in V: R(T) = {T(x) : x ∈ V}

Theorem 2.1

For vector spaces V , W and linear T : V → W, N (T ) and R (T ) are subspaces of V and W, respectively.

Theorem 2.2For vector spaces V , W and linear T : V → W, if β = {v1, . . . , vn}is a basis for V, then

R (T ) = span(T (β )) = span({T (v1), . . . , T (vn)})



Nullity and Rank

Definition

For vector spaces V, W and linear T : V → W, if N(T) and R(T)are finite-dimensional, the nullity and the rank of T are thedimensions of N(T) and R(T), respectively.

Theorem 2.3 (Dimension Theorem)

For vector spaces V , W and linear T : V → W, if V is finite-dimensional then

nullity(T ) + rank(T ) = dim(V )



Properties of Linear Transformations

Theorem 2.4

For vector spaces V , W and linear T : V → W, T is one-to-one if and only if N (T ) = {0 }.

Theorem 2.5For vector spaces V , W of equal (finite) dimension and linear T : V → W, the following are equivalent:

(a) T is one-to-one

(b) T is onto (c) rank(T ) = dim(V )



Linear Transformations and Bases

Theorem 2.6

Let V , W be vector spaces over F and {v1, . . . , vn} a basis for V.For w1, . . . , wn in W, there exist exactly one linear transformation

T : V → W such that T (vi) = wi for = 1, . . . , n.

Corollary

Suppose {v1, . . . , vn} is a finite basis for V, then if U , T : V → W

are linear and U (vi) = T (vi) for i = 1, . . . , n, then U = T.



Coordinate Vectors

Definition

For a finite-dimensional vector space V, an ordered basis for V is abasis for V with a specific order. In other words, it is a finitesequence of linearly independent vectors in V that generates V.

Definition

Let β = {u1, . . . , un} be an ordered basis for V, and for x ∈ V leta1, . . . , an be the unique scalars such that

x =n

i=1aiui.

The coordinate vector of x relative to β is

[x]β =

a1...

an



Matrix Representations

Definition

Suppose V, W are finite-dimensional vector spaces with orderedbases β = {v1, . . . , vn}, γ = {w1, . . . , wm}. For linear T : V → W,there are unique scalars aij ∈ F such that

T(v j) =

mi=1

aijwi for 1 ≤ j ≤ n.

The m × n matrix A defined by Aij = aij is the matrix representation of T in the ordered bases β and γ , written

A = [T]γ

β. If V = W and β = γ , then A = [T]β .

Note that the jth column of A is [T(v j)]γ , and if [U]γ β = [T]γ β forlinear U : V → W, then U = T.



Addition and Scalar Multiplication

Definition

Let T, U : V → W be arbitrary functions of vector spaces V, W over

F . Then T + U, aT : V → W are defined by(T + U)(x) = T(x) + U(x) and (aT)(x) = aT(x), respectively, forall x ∈ V and a ∈ F .

Theorem 2.7

With the operations defined above, for vector spaces V , W over F

and linear T , U : V → W:

(a) aT + U is linear for all a ∈ F

(b) The collection of all linear transformations from V to W is a

vector space over F

Definition

For vector spaces V, W over F , the vector space of all lineartransformations from V into W is denoted by L(V, W), or justL(V) if V = W.

M i R i



Matrix Representations

Theorem 2.8

For finite-dimensional vector spaces V , W with ordered bases β, γ ,

and linear transformations T , U : V → W:(a) [T + U ]γ β = [T ]γ β + [U ]γ β

(b) [aT ]γ β = a[T ]γ β for all scalars a

C i i f Li T f i



Composition of Linear Transformations

Theorem 2.9

Let V , W , Z be vector spaces over a field F , and T : V → W,U : W → Z be linear. Then UT : V → Z is linear.

Theorem 2.10Let V be a vector space and T , U 1, U 2 ∈ L(V ). Then

(a) T (U 1 + U 2) = TU 1 + TU 2 and (U 1 + U 2)T = U 1T + U 2T

(b) T (U 1U 2) = (TU 1)U 2

(c) TI = IT = T (d) a(U 1U 2) = (aU 1)U 2 = U 1(aU 2) for all scalars a

M i M l i li i



Matrix Multiplication

Let T : V → W, U : W → Z, be linear, α = {v1, . . . , vn},

β = {w1, . . . , wm}, γ = {z1, . . . , z p} ordered bases for U, W, Z,and A = [U ]γ β, B = [T ]βα. Consider [UT]γ α:

(UT)(v j) = U(T(v j)) = U

m

k=1Bkjwk

=

m

k=1BkjU(wk)

=mk=1

Bkj

pi=1

Aikzi

=

pi=1

mk=1

AikBkj

zi

Definition

Let A,B be m × n, n × p matrices. The product AB is the m × p

matrix with

(AB)ij =n

k=1AikBkj , for 1 ≤ i ≤ m, 1 ≤ j ≤ p

M t i M lti li ti



Matrix Multiplication

Theorem 2.11

Let V, W, Z be finite-dimensional vector spaces with ordered bases α,β,γ , and T : V → W, U : W → Z be linear. Then

[UT ]γ α = [U ]γ β[T ]βα

Corollary

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

Definition

The Kronecker delta is defined by δ ij = 1 if i = j and δ ij = 0 if i = j . The n × n identity matrix I n is defined by (I n)ij = δ ij .

P ti



Properties

Theorem 2.12

Let A be m × n matrix, B,C be n × p matrices, and D, E be q × m 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) I mA = A = AI n

(d) If V is an n-dimensional vector space with ordered basis β ,then [I V ]β = I n

Corollary

Let A be m × n matrix, B1, . . . , Bk be n × p matrices, C 1, . . . , C kbe q × m matrices, and a1, . . . , ak be scalars. Then

A k

i=1

aiBi =k

i=1

aiABi and k

i=1

aiC iA =k

i=1

aiC iA

Properties



Properties

Theorem 2.13Let A be m × n matrix and B be n × p matrix, and u j, v j the jthcolumns of AB,B. Then

(a) u j = Av j

(b) v j = Be j

Theorem 2.14

Let V, W be finite-dimensional vector spaces with ordered bases

β, γ , and T : V → W be linear. Then for u ∈ V:

[T (u)]γ = [T ]γ β [u]β

Left multiplication Transformations



Left-multiplication Transformations

Definition

Let A be m × n matrix. The left-multiplication transformation LAis the mapping LA : Fn → Fm defined by LA(x) = Ax for eachcolumn vector x ∈ Fn.

Theorem 2.15

Let A be m × n matrix, then LA : F n

→ F m

is linear, and if B is m × n matrix and β, γ are standard ordered bases for F n, F m, then:

(a) [LA]γ β = A

(b) LA = LB if and only if A = B

(c) LA+B = LA + LB and LaA = aLA for all a ∈ F (d) For linear T : F n → F m, there exists a unique m × n matrix C

such that T = LC , and C = [T ]γ β(e) If E is an n × p matrix, then LAE = LALE

(f) If m = n then LI n

= I F n

Associativity of Matrix Multiplication



Associativity of Matrix Multiplication

Theorem 2.16

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

Inverse of Linear Transformations



Inverse of Linear Transformations

Definition

Let V, W be vector spaces and T : V → W be linear. A function U: W → V is an inverse of T if TU=IW and UT=IV. If T has aninverse, it is invertible and the inverse T−1 is unique.

For invertible T,U:

1 (TU)−1 = U−1T−1

2 (T−1)−1 = T (so T−1 is invertible)

3 If V,W have equal dimensions, linear T : V → W is invertibleif and only if rank(T) = dim(V)

Theorem 2.17

For vector spaces V,W and linear and invertible T : V → W,T −1 : W → V is linear.

Inverses



Inverses

DefinitionAn n × n matrix A is invertible if there exists an n × n matrix B

such that AB = BA = I .

Lemma

For invertible and linear T from V to W, V is finite-dimensional if and only if W is finite-dimensional. Then dim(V ) = dim(W ).

Theorem 2.18

Let V,W be finite-dimensional vector spaces with ordered bases β, γ , and T : V → W be linear. Then T is invertible if and only if [T ]γ β is invertible, and [T −1]βγ = ([T ]γ β)−1.

Inverses



Inverses

Corollary 1

For finite-dimensional vector space V with ordered basis β and linear T : V → V, T is invertible if and only if [T ]β is invertible,

and [T −1

]β = ([T β ])−1

.

Corollary 2

An n × n matrix A is invertible if and only if LA is invertible, and (L

A)−1 = L

A−1 .

Isomorphisms



Isomorphisms

DefinitionLet V,W be vector spaces. V is isomorphic to W if there exists alinear transformation T : V → W that is invertible. Such a T is anisomorphism from V onto W.

Theorem 2.19

For finite-dimensional vector spaces V,W, V is isomorphic to W if and only if dim(V ) = dim(W ).

CorollaryA vector space V over F is isomorphic to F n if and only if dim(V ) = n.

Linear Transformations and Matrices



Linear Transformations and Matrices

Theorem 2.20

Let V,W be finite-dimensional vector spaces over F of dimensions

n,m with ordered bases β, γ . Then the functionΦ : L(V , W ) → M m×n(F ), defined by Φ(T ) = [T ]γ

β

for T ∈ L(V , W ), is an isomorphism.

Corollary

For finite-dimensional vector spaces V,W of dimensions n,m,

L(V , W ) is finite-dimensional of dimension mn.

The Standard Representation



The Standard Representation

Definition

Let β be an ordered basis for an n-dimensional vector space V overthe field F . The standard representation of V with respect to β is

the function φβ : V → Fn

defined by φβ(x) = [x]β for each x ∈ V.

Theorem 2.21

For any finite-dimensional vector space V with ordered basis β , φβis an isomorphism.

The Change of Coordinate Matrix



The Change of Coordinate Matrix

Theorem 2.22Let β and β be ordered bases for a finite-dimensional vector space V, and let Q = [I V ]

ββ. Then

(a) Q is invertible

(b) For any v ∈ V, [v]β = Q[v]β

Q = [IV]ββ is called a change of coordinate matrix , and we say thatQ changes β -coordinates into β -coordinates .

Note that if Q changes from β into β coordinates, then Q−1

changes from β into β coordinates.



Linear Functionals



A linear functional on a vector space V is a linear transformationfrom V into its field of scalars F .

Example

Let V be the continuous real-valued functions on [0, 2π]. For a fix

g ∈ V, a linear functional h : V → R is given by

h(x) = 1

2π

2π

0

x(t)g(t) dt

Example

Let V = Mn×n(F ), then f : V → F with f (A) = tr(A) is a linearfunctional.

Coordinate Functions



Example

Let β = {x1, . . . , xn} be a basis for a finite-dimensional vectorspace V. Define f i(x) = ai, where

[x]β =a1...

an

is the coordinate vector of x relative to β . Then f i is a linear

functional on V called the ith coordinate function with respect to the basis β . Note that f i(x j) = δ ij .

Dual Spaces



p

Definition

For a vector space V over F , the dual space of V is the vectorspace V∗ = L(V, F ).

Note that for finite-dimensional V,

dim(V∗) = dim(L(V, F )) = dim(V) · dim(F ) = dim(V)

so V and V∗ are isomorphic. Also, the double dual V∗∗ of V is thedual of V∗.

Dual Bases



Theorem 2.24

Let β = {x1, . . . , xn} be an ordered basis for finite-dimensional

vector space V, and let f i be the ith coordinate function w.r.t. β ,and β ∗ = {f 1, . . . , f n}. Then β ∗ is an ordered basis for V ∗ and for any f ∈ V ∗,

f =n

i=1f (xi)f i.

Definition

The ordered basis β ∗ = {f 1, . . . , f n} of V∗ that satisfies f i(x j) = δ ijis called the dual basis of β .

Theorem 2.25

Let V , W be finite-dimensional vector spaces over F with ordered bases β, γ . For any linear T : V → W, the mapping T t : W ∗ → V ∗

defined by T t(g ) = gT for all g ∈ W ∗ is linear with the property

[T t

]β∗

γ ∗ = ([T γ β)

t

.

Double Dual Isomorphism



For a vector x ∈ V, define x̂ : V∗ → F by x̂(f ) = f (x) for every

f ∈ V∗

. Note that x̂ is a linear functional on V∗

, so x̂ ∈ V∗∗

.

Lemma

For finite-dimensional vector space V and x ∈ V, if x̂(f ) = 0 for all f ∈ V ∗, then x = 0.

Theorem 2.26

Let V be a finite-dimensional vector space, and define ψ : V → V ∗∗

by ψ(x) = x̂. Then ψ is an isomorphism.

Corollary

For finite-dimensional V with dual space V ∗, every ordered basis for V ∗ is the dual basis for some basis for V.