전체 글

(1)

Vector Space

Wanho Choi

(wanochoi.com)

(2)

Linear Combination of Vectors

Combining

m n-dimensional vectors

using

scalar multiplications

and

vector additions

y

= a

1

x

1

+ a

2

x

2

+ a

3

x

3

+!+ a

m

x

m

=

a

i

x

i i=1 m

a

i

∈!, x

i

∈!

n

, m

∈"

(

)

(3)

Vector Space

A

n-dimensional space

in which any

linear

combination

of a set of vectors is also in that

space

y

= a

1

x

1

+ a

2

x

2

+ a

3

x

3

+!+ a

m

x

m

=

a

i

x

i i=1 m

y

∈!

n

a

i

∈!, x

i

∈!

n

, m

∈"

(

)

(4)

Linearly Dependent / Independent

: linearly dependent

Otherwise, they are

linearly independent

.

x

1

,x

2

,x

3

,

!,x

m

iff a

1

,a

2

,a

3

,

!,a

m

exist for

a

1

x

1

+ a

2

x

2

+ a

3

x

3

+!+ a

m

x

m

=

a

i

x

i i=1 m

= 0

a

1

× a

2

× a

3

×!× a

m

≠ 0

(

)

(5)

Basis

A set of n vectors

in a

n-dimensional space

,

which are

linearly independent

and every vector

in that space can be reproduced by a

linear

combination

of this set.

a

1

e

1

+ a

2

e

2

+ a

3

e

3

+!+ a

n

e

n

∈!

n

a

i

∈!, e

i

∈!

n

(6)

Orthonormal Basis

A set of n vectors

in a

n-dimensional space

,

which are

linearly independent

and every vector

in that space can be reproduced by a

linear

combination

of this set.

a

1

e

1

+ a

2

e

2

+ a

3

e

3

+!+ a

n

e

n

∈!

n

a

i

∈!, e

i

∈!

n

(

)

e

i

⋅e

j

=

0 (i

≠ j)

1 (i

= j)

⎩⎪

(7)

Span

All

linear combination

of

basis vectors

a

i

∈!, e

i

∈!

n

(

)

(8)

Dimension & Rank

A

m×n

maps

vectors in

n

to vectors in

m

.

The

row space

of this matrix is the vector space

generated by linear combinations of the row

vectors.

The

rank

of this matrix is

the maximum number

(9)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

(10)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

(11)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

r : the row rank of A

= n A m r B Cn r m

(12)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

r : the row rank of A

= n A m r B Cn r m

(13)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

a11 a12 a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x b2 x b1y b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ c1x c1y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ a11 a12 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x c1x c1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ b2 x c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 1st row vector a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1y c1x c1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ b2 y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 2nd row vector

r : the row rank of A

= n A m r B Cn r m

(14)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

a11 a12 a21 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1x b2 x b1y b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ c1x c1y c2 x c2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1xc1y + b2 xc2 y b1yc1x + b2 yc2 x b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ a11 a21 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1x + b2 xc2 x b1yc1x + b2 yc2 x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = c1x b1x b1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ c2 x b2 x b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 1st column vector a12 a22 ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = b1xc1y + b2 xc2 y b1yc1y + b2 yc2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = c1y b1x b1y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥+ c2 y b2 x b2 y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥

⎥: the 2nd column vector

r : the row rank of A

= n A m r B Cn r m

(15)

Row Rank = Column Rank

A

m

×n

= B

m

×r

C

r

×n

=

the row space of A is the span of the rows of C

the column space of A is the span of the columns of B

r : the row rank of A

m n A m r B Cn r

the row rank of A = the number of independent row vectors of C = r

the column rank of A = the number of independent column vectors of B = r

(16)

Row / Column Space

a

11

a

12

! a

1n

a

21

a

22

! a

2n

!

!

"

!

a

m1

a

m 2

! a

mn

A

m

×n

a

11

a

12

! a

1n

a

21

a

22

! a

2n

!

!

"

!

a

m1

a

m 2

! a

mn

Row Space

: the subspace of !n spanned by the row vectors of A

row vectors

∈!

n

column vectors

∈!

m

Column Space

(17)

a

11

a

12

! a

1n

a

21

a

22

! a

2n

!

!

"

!

a

m1

a

m 2

! a

mn

x

1

x

2

!

x

n

=

b

1

b

2

!

b

m

Ax

= b

m× n n×1 m×1

x

1

a

11

a

21

!

a

m1

+ x

2

a

12

a

22

!

a

m 2

+!+ x

n

a

1n

a

2n

!

a

mn

=

b

1

b

2

!

b

m

Row / Column Space

The solution (x

1

, x

2

,

!, x

n

) exists

iff b is in the column space of A.

(18)

Ax

= 0

m× n n×1 m×1

Null Space

Null Space

: the set of vectors x such that Ax=0.

(19)

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