Matrices, Vectors and Linear Combinations: Lecture 1 of MATH 250 Summer 2022, Exercises of Linear Algebra

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Lecture 1

Matrices, Vectors and Linear Combinations

MATH 250 SUMMER 2022

May 31, 2022

Matrices

Definition (Matrix)

A matrix is a rectangular array of real numbers.

Size If a matrix has m rows and n columns, then we say the

matrix is m ⇥ n (m by n).

Entry Let 1  i  m and 1  j  n. We say the (i, j)-entry of a

m ⇥ n matrix is the real number in the ith row and the jth

column.

Notation: We usually denote matrices using uppercase letters,

e.g.A, B, C ... and their (i, j)-entries using lowercase letters, e.g.aij , bij , cij ....

row

index

-4,43s ]

in

↑

[

column

index

§

TAKI

←

A

22=

Relations and Actions on Mm⇥n(R)

Definition

Consider m ⇥ n matrices A, B, C and a real number s, i.e.

A, B, C 2 Mm⇥n(R), s 2 R.

Equality We say A = B if aij = bij , for any i, j.

Addition We say C = A + B if cij = aij + bij , for any i, j.

Scalar Multiplication We say C = sA if cij = saij , for any i, j.

Tranpose We say D = A

T if dij = aji , for any i, j.

Question: what is the size of D? is D also a m ⇥ n matrix?

AÉ

' '

7,13=-417-1--

213

• -^ - -^ ☒

nxm

a. (^) range

of

'z A- c- (^) Mmxn

≤n DEM nxm

Example 2

Determine whether A = B. Compute 2A B, B

T .

43T¥

=

By:B

BE

2 × 3 'aÉB%ÉÉ^ #

2A

• B

É¥E]

Ii¥Ñf

III.

Example 3

Given A, write out a 2. Compute u + v.

3

column

vectors

A!

E)

ED

TH

IE

]

,

Ñz

És

• (^) -

ai

• ii. =^

[

E)

=

-4 (^) ]

8 Properties of Mm⇥n(R) and R

m

Here we use x, y , z to represent matrices or vectors and s, t to represent

real numbers. There are 8 key properties. 1-4 for addition, 5-6 for scalar

multiplication and 7-8 for interaction between + and scalar multiplication.

Addtion is commutative x + y = y + x.

Addtion is associative (x + y ) + z = x + (y + z).

0 is neutral 0 + x = x.

Inverse exists x + x = 0.

Scaling by 1 1 x = x.

Scaling is associative s(tx) = (st)x.

Distribution law 1 (s + t)x = sx + tx.

Distribution law 2 s(x + y ) = sx + sy.

• €m (^) _m€

}

"%¥÷

¥

A

=

⑤

④]zx

}

E¥¥%

-1%+5151+

"

¥¥

' = (^) $+6'D

0

=

'

Example 4

Compute Av. Prove the first property.

Proved

AiY=?Añ+A

Ñ+P=μ

.

Untvn

it's

-- ¥¥¥÷¥"¥_

-+

tuiaitvnñn

= A-u→+A=RHS

Matrix

vector multi^

(¥E

÷

:* Ed

ed

if

]

-151 :]

=

F)

E)

=

R

?

e 51

R

?

E-

Special Vectors and Matrices

The standard vectors in R

n : e 1 , e 2 , · · · , en. (We have property

Aej = aj .)

The identity n ⇥ n matrix: In. (We have property Inv = v.)

The rotation 2 ⇥ 2 matrices: A✓.

n

I

diagonal entry

ei .

éi

E.

=

-4¥

_

'

¥ ]

Et

'

Aa

-1¥

. ]l¥±

A É=E¥÷=E¥

Example 5

Compute A ⇡ 4

e 1.

'

*f¥:¥÷;¥!¥¥¥?

☒-

? What iseIinR

}

Attqeiyyo

Homework 1, part A

Solve problems 3, 5, 17, 25, 28 from Section 1.1, on page 11.

Solve problems 1, 9, 15, 17, 35, 39 from Section 1.2, on page 25.

Solve problems 66, 76 from Section 1.2, on page 26.