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İ S. Y

M ,

Surname, Name:

Q . §. S Rn

Let Abe the 4×5matrix

5 2 0 −8−8

4 1 2 −8−9

5 1 3 5 19

−8−5 6 8 5

.

A) Find the null space of A.

B) Find the column space of A.

C) Find a basis for the null space of A.

D) Find a basis for the column space of A.

A .

Before we answer the parts in the question, we need to reduce the given

matrix. We see that

5 2 0 −8−8

4 1 2 −8−9

5 1 3 5 19

−8−5 6 8 5

∼

1 0 0 60 122

010−154 −309

001 −47 −94

0 0 0 0 0

.

(A) We need to solve Ax= 0. Note that x1, x2, x3are basic, x4and x5are free

variables. Therefore, we see that

Nul A =

x4

−60

154

47

1

0

+x5

−122

309

94

0

1

:x4, x5∈R

which is a subspace of R5.

(B) Col A is the set of all linear combinations of the columns of A. Therefore,

we see that

Col A =

c1

5

4

5

−8

+c2

2

1

1

−5

+c3

0

2

3

6

+c4

−8

−8

5

8

+c5

−8

−9

19

5

:x4, x5∈R

1

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