Lecture 1 Matrix Terminology and Notation, Lecture notes of Linear Algebra

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Lecture 1

Matrix Terminology and Notation

matrix dimensions

column and row vectors

special matrices and vectors

Matrix dimensions

a

matrix

is a rectangular array of numbers between brackets

examples:

A

.^3

.^1

.^3

.^1

.^1

.^7

B

[

]

dimension

(or size) always given as (numbers of) rows

×

columns

A

is a

×

matrix,

B

is

×

the matrix

A

has four columns;

B

has two rows

m

×

n

matrix is called

square

if

m

n

,^

fat

if

m < n

,^

skinny

if

m > n

Matrix Terminology and Notation

Column and row vectors

a matrix with one column,

i.e.

, size

n

×

, is called a (column)

vector

a matrix with one row,

i.e.

, size

×

n

, is called a

row vector

‘vector’ alone usually refers to column vectorwe give only one index for column & row vectors and call entries components

v

.^3

.^3

w

[

.^1

]

v

is a

-vector (or

×

matrix); its third component is

v

3

.^3

w

is a row vector (or

×

matrix); its third component is

w

3

Matrix Terminology and Notation

Matrix equality

A

B

means:

A

and

B

have the same size

the corresponding entries are equal

for example,^ •

[

.^3

]

[

.^3

]

since the dimensions don’t agree

[

.^3

]

[

.^1

]

since the 2nd components don’t agree

Matrix Terminology and Notation

Unit vectors

e

i^

denotes the

i

th

unit vector

: its

i

th component is one, all others zero

the three unit

-vectors are: e

1

e

2

e

3

as usual, you have to figure the size out from contextunit vectors are the columns of the identity matrix

I

some authors use

(or

e

) to denote a vector with all entries one,

sometimes called the

ones vector

the ones vector of dimension

is

[

]

Matrix Terminology and Notation