Norman and Wolczuk Introduction to Linear Algebra for ... · PDF fileNorman and Wolczuk Introduction to Linear Algebra for Science and Engineering Author: Chapter 3: Matrices, Linear

Norman and WolczukIntroduction to Linear Algebrafor Science and Engineering

Chapter 3: Matrices, Linear Mappings, and Inverses

Copyright c©2012 Pearson Canada Inc. 3-1

Matrices (§3.1)An m × n matrix has m row and n columns:

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

The entry in row i and column j of a matrix A is denoted (A)ij .

Two matrices A and B are equal if (A)ij = (B)ij for 1 ≤ i ≤ m,1 ≤ j ≤ n.

A matrix is square if it is n × n.

A matrix is upper triangular if the entries beneath the main diagonalare all zero.

A matrix is lower triangular if the entries above the main diagonal areall zero.

A matrix is diagonal if is both upper and lower triangular.


Operations on Matrices (§3.1)

Suppose A and B are m × n matrices and t ∈ R. We define addition ofmatrices by

(A + B)ij = (A)ij + (B)ij ,

and scalar multiplication by

(tA)ij = t(A)ij .


The Transpose of a Matrix (§3.1)

Definition (Transpose)

Let A be an m × n matrix. The transpose of A is the n ×m matrix,denoted AT , such that

(AT )ij = (A)ji .

Properties of the Transpose

For any matrices A and B (of the same size) and s ∈ R, we have

1 (AT )T = A,

2 (A + B)T = AT + BT ,

3 (sA)T = sAT .


Matrix Multiplication (§3.1)

Summation Notationn∑

k=1

ak = a1 + a2 + · · ·+ an

Definition (Matrix Multiplication)

Let B be an m× n matrix with rows ~bT1 , . . . ,~bTm and A be an n× p matrix

with columns ~a1, . . . ,~ap. Then BA is the m × p matrix with

(BA)ij = ~bi ·~aj =n∑

k=1

(A)ik(B)kj .


Matrix Multiplication (§3.1)

Theorem

If A, B, and C are matrices of the correct size so that the requiredproducts are defined, and t ∈ R, then

1 A(B + C ) = AB + AC ,

2 t(AB) = (tA)B + A(tB),

3 A(BC ) = (AB)C ,

4 (AB)T = BTAT .

Warnings

1 The matrix product is not commutative. In general, AB 6= BA.

2 In general, AB = AC does not imply that B = C .

Theorem

If A and B are m × n matrices such that A~x = B~x for all x ∈ Rn, thenA = B.


Identity Matrix (§3.1)

Definition (Identity Matrix)

The n × n matrix

In = diag(1, 1, . . . , 1) =

1 0 · · · 0

0 1. . .

......

. . .. . . 0

0 · · · 0 1

is called the identity matrix.

Theorem

If A is any m × n matrix, then ImA = A = AIn.


Matrix Mappings (§3.2)

Definition (Matrix Mapping)

For an m × n matrix A, the matrix mapping corresponding to A is thefunction

fA : Rn → Rm, fA(~x) = A~x , ~x ∈ Rn.

Theorem

Let ~e1,~e2, . . . ,~en be the standard basis vectors of Rn and let A be anm × n matrix. Then, for ~x ∈ Rn, we have

fA(~x) = x1fA(~e1) + x2fA(~e2) + · · ·+ xf fA(~en).

Theorem

Let A be an m × n matrix. Then, for any ~x , ~y ∈ Rn and t ∈ R, we have

(L1) fA(~x + ~y) = fA(~x) + fA(~y),

(L2) fA(t~x) = tfA(~x).


Linear Mappings (§3.2)

Definition (Linear Mapping)

A function L : Rn → Rm is a linear mapping (or linear transformation) if,for every ~x , ~y ∈ Rn and t ∈ R, it satisfies

(L1) L(~x + ~y) = L(~x) + L(~y),

(L2) L(t~x) = tL(~x).

Definition (Linear Operator)

A linear operator is a linear mapping whose domain and codomain are thesame.

Theorem

If L : Rn → Rm is a linear mapping, then L can be represented as a matrixmapping, with the corresponding m × n standard matrix

[L] =[L(~e1) L(~e2) · · · L(~en)

].


Compositions and Linear Combinationsof Linear Mappings (§3.2)Suppose L,M : Rn → Rm are linear mappings and t ∈ R.

Definition (Operations on Linear Mappings)

We define (L + M) and (tL) to be the mappings from Rn to Rm given by

(L + M)(~x) = L(~x) + M(~x),

(tL)(~x) = tL(~x).

Theorem

The mappings (L + M) and (tL) are linear mappings.

Definition (Composition of Linear Mappings)

Let N : Rm → Rp be another linear mapping. The composition of N and Lis the (linear) mapping N ◦ L : Rn → Rp defined by

(N ◦ L)(~x) = N(L(~x)), x ∈ Rn.


Compositions and Linear Combinationsof Linear Mappings (§3.2)

Theorem

Let L : Rn → Rm, M : Rn → Rm, and N : Rm → Rp be linear mappingsand t ∈ R. Then

[L + M] = [L] + [M], [tL] = t[L], [N ◦ L] = [N][L].

Definition (Identity Mapping)

The identity mapping is the linear mapping Id : Rn → Rn defined by

Id(~x) = ~x .


Rotations in the Plane (§3.3)Rθ : R2 → R2 is defined to be the transformation that rotates ~xcounterclockwise through angle θ to the image Rθ(~x). This is a linearmapping and

[Rθ] =

[cos θ − sin θsin θ cos θ

].

x1

x2

θ

�x

Rθ(�x)


Rotation Through Angle θ Aboutthe x3-axis in R3 (§3.3)If R denotes the a right-handed counterclockwise rotation through anangle θ about the x3-axis in R3, then

[R] =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

.

x1

x2

x3

θ

θ

(1, 0, 0)

(1, 0, x3)

(0, 0, x3)

(cos θ, sin θ, 0)

(cos θ, sin θ, x3)


Stretches, Contractions, and Dilations (§3.3)Stretches

The linear transformation with matrix[t 00 1

], t > 0,

is called a shrink (in the x1-direction).

Contractions and Dilations

Consider the linear operator T : R2 → R2 with matrix[t 00 t

], t > 0.

Thus, T (~x) = t~x for all x ∈ R2.

If 0 < t < 1, T is called a contraction.

If t > 1, T is called a dilation.


Shears (§3.3)

A linear transformation S : R2 → R2 with matrix[1 s0 1

]is called a shear in the direction of x1 by amount s.

x1

x2

(0, 1)(2, 1)

(2, 0)

(5, 1) (2 + 5, 1)


Reflections (§3.3)Consider a line in R2 or a plane in R3 with equation ~n · ~x = 0. Thereflection in the line/plane with normal vector ~n is the linear mapping

refl~n(~p) = ~p − 2 proj~n(~p).

x1

x2

�a

refl�n �aθ

�n

θ

proj�n �a


Solution Space and Nullspace (§3.4)

Theorem/Definition

Let A be an m × n matrix. The set

S = {~x ∈ Rn | A~x = ~0}

of all solutions to a homogeneous system A~x = ~0 is a subspace of Rn. It iscalled the solution space of the system.

Definition (Nullspace)

The nullspace (or kernel) of a linear mapping L : Rn → Rm is the set

Null(L) = {~x ∈ Rn | L(~x) = ~0}.

The nullspace of an m × n matrix A is

Null(A) = {~x ∈ Rn | A~x = ~0}.


Solution Set of A~x = ~b (§3.4)

Theorem

Let ~p be a solution of the system A~x = ~b, ~b 6= ~0.

1 If ~v is any other solution of the same system, then A(~p − ~v) = ~0.

2 If ~h is any solution of the corresponding system A~x = ~0, then ~p + ~h isa solution of the system A~x = ~b.

The solution ~p above is sometimes called a particular solution of thesystem.


Range of L and Columnspace of A (§3.4)

Definition (Range)

The range of a linear mapping L : Rn → Rm is the set

Range(L) = {L(~x) ∈ Rm | x ∈ Rn}.

Definition (Columnspace)

The columnspace of an m × n matrix A is the set

Col(A) = {A~x ∈ Rm | x ∈ Rn}.

If L : Rn → Rm is a linear mapping, then

Range(L) = Col([L]).

Theorem

The system A~x = ~b is consistent if and only if ~b ∈ Col(A).


Rowspace of a Matrix and a Basisof the Rowspace (§3.4)

Definition (Rowspace)

The rowspace of an m × n matrix A is the subspace of Rn spanned by therows of A (regarded as vectors) and is denoted Row(A).

Theorem

If a matrix A is row equivalent to another matrix B, thenRow(A) = Row(B).

Theorem

Let B be the reduced row echelon form of a matrix A. Then the non-zerorows of B form a basis for Row(A), and hence the dimension of Row(A)equals the rank of A.


Basis of the Columnspace and Nullspaceof a Matrix (§3.4)

Theorem

Suppose that B is the reduced echelon form of A. Then the columns of Athat correspond to the columns of B with leading 1s form a basis of thecolumnspace of A. Hence, the dimension of the columnspace equals therank of A.

Definition (Nullity)

The dimension of the nullspace of a matrix A is the nullity of A and isdenoted by nullity(A).

Theorem

Let A be an m × n matrix with rank(A) = r . Then the spanning set forthe general solution of the homogeneous system A~x = ~0 obtained by themethod in Chapter 2 is a basis for Null(A) and the nullity of A is n − r .


Rank Theorem (§3.4)

Rank Theorem

If A is an m × n matrix, then

rank(A) + nullity(A) = n.

A Summary of Facts About Rank

For an m × n matrix A, rank(A) is equal to the following:

the number of leading 1s in the reduced echelon form of A,

the number of non-zero rows in any row echelon form of A,

dim Row(A),

dim Col(A),

n − dim Null(A).


Inverses (§3.5)

Definition (Inverse)

Let A be an n × n matrix. If there exists an n × n matrix B such thatBA = AB = I , then A is invertible, and B is the inverse of A (and A is theinverse of B). The inverse of A is denoted A−1.

Theorem

Suppose that A and B are n × n matrices such that AB = I . ThenBA = I , so that B = A−1. Moreover, B and A have rank n.

Theorem

Suppose A and B are invertible matrices and t is a non-zero real number.

(tA)−1 = 1tA

−1

(AB)−1 = B−1A−1

(AT )−1 = (A−1)T


Finding the Inverse of a Matrix (§3.5)

Algorithm

To find the inverse of a square matrix A:

1 Row reduce the multi-augmented matrix[A I

]so that the left

block is in reduced row echelon form.

2 If the reduced row echelon form is[I B

], then A−1 = B.

3 If the reduced row echelon form of A is not I , then A is not invertible.


Inverse Linear Mappings (§3.5)

Definition (Inverse Mapping)

If L : Rn → Rm is a linear mapping and there exists another linearmapping M : Rn → Rm such that M ◦ L = Id = L ◦M, then L is said to beinvertible, and M is called the inverse of L, denoted L−1.

Theorem

Suppose L : Rn → Rm and M : Rn → Rm are linear mappings. Then M isthe inverse of L if and only if [M] is the inverse of [L].


Invertible Matrix Theorem (§3.5)

Theorem (Invertible Matrix Theorem)

Suppose L : Rn → Rn is a linear mapping with standard (n × n) matrix A.Then the following statements are equivalent.

1 A is invertible.

2 rank(A) = n.

3 The reduced row echelon form of A is I .

4 For all ~b ∈ Rn, the system A~x = ~b is consistent and has a uniquesolution.

5 The columns of A are linearly independent.

6 The columnspace of A is Rn.

7 L is invertible.

8 Range(L) = Rn.

9 Null(L) = {~0}.


Elementary Matrices (§3.6)

Definition (Elementary Matrix)

A matrix that can be obtained from the identity matrix by a singleelementary row operation is called an elementary matrix.

Theorem

If A is an n× n matrix and E is the elementary matrix obtained from In bya certain elementary row operation, then the product EA is the matrixobtained from A by the same elementary row operation.

Theorem

For any matrix A, there exists a sequence of elementary matrices,E1,E2, . . . ,Ek , such that Ek · · ·E2E1A is equal to the reduced row echelonform of A.

Theorem

If an n× n matrix A has rank n, then it can be represented as a product ofelementary matrices.


LU-Decomposition (§3.7)

Theorem/Definition

If A is an n × n matrix that can be row reduced to row echelon formwithout swapping rows, then there exists an upper triangular matrix U andlower triangular matrix L such that A = LU. This is called anLU-decomposition of A.

Solving Systems with the LU-Decomposition (§3.7)

If A = LU, with U upper triangular and L lower triangular, the systemA~x = ~b can be rewritten as

LU~x = ~b.

Letting ~y = U~x , we have

L~y = ~b and U~x = ~y .

Since both L and U are triangular, we can solve both systems immediately,using substitution.


Norman and Wolczuk Introduction to Linear Algebra for ... · PDF fileNorman and Wolczuk Introduction to Linear Algebra for Science and Engineering Author: Chapter 3: Matrices, Linear

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