- 1 -

Chapter 5

5.1

5.1.1 ,

(1) , (2) , , (linear transformation) (linear map) . , (linear operator) .

- 2 -

5.1.2 .

,

. , .

5.1.3 ,

. 5.1.4

. (formal derivative) .

5.1.4 .

- 3 -

.

, 2.1.5

. .

5.1.5 ,

.

,

.

,

- 4 -

. ,

.

5.1.4 5.1.5

.

( ) .

5.1.6 ,

.

.

.

,

5.1.7 . (1)

- 5 -

(2) (3) (1)

4.1.7 (1) .(2) (3)

5.1.8 ,

. ,

. .

5.1.9 , , .

- 6 -

5.2

5.2.1 ,

(Kernel) .

(Range) .

5.2.2 , , (1) . (2) .(1)

,

. 4.2.2 .(2) .

. ,

- 7 -

. 4.2.2 .

5.2.3 ,

, (column space).

5.2.4 , nullity nullity . rank rank .

5.2.5 .nullity rank dim

dim nullity .

,

. ,

. , . ,

. ,

- 8 -

. ,

.

. .

, .

.

. . rank dim , nullity .

5.2.6 ,

,

. .

- 9 -

,

.

. nullity dim .

(= ).

. rank dim . ,

llity rank dim .

- 10 -

5.3

5.3.1

, ( , one to one) .

5.3.2 . .

()

. , .() , .

. , .

5.3.3 ,

. , .

, . .

. ,

- 11 -

. .

. .

5.3.4 , , (isomorphism) . , (isomorphic) .

, .

5.3.5 , . ,

. , ,

, .

5.3.4 4 ,

. .

- 12 -

5.3.6 .(1) . .

(2) . .(3) . .

- 13 -

5.4

,

(ordered basis) .

.

.

(isomorphism).

(coordinate

vector) .

.

- 14 -

.

.

,

.

.

,

.

. .

- 15 -

,

.

. .

.

. .

.

5.4.1 ,

.

. .

- 16 -

, , . ,

,

.

- 17 -

5.5

.

.

,

. ,

. (transition matrix) .

.

,

.

- 18 -

,

.

,

.

,

.

.

5.5.1

.

.

.