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Spaningset theorem

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Spaning set theorem (new Question)

Hi

I have a couple of questions regarding an assignment which deals with the spanning set theorem.

Hope You can help

The matrix [tex]A = [a1 \ a2 \ a3 \ a4 \ a5] = \left[ \begin{array}{ccccc} 1 & 0 & 0 & 1 & -1 \\ 0 & 1 & 1 & 2 & 0 \\ 1 & 0 & 0 & 1 & 2 \\ -1 & 2 & 2 & 3 & 1 \end{array} \right][/tex]

a) First I determin the rank of A rank(A) = 3

The dimension of Null A: dim (Null A) = 2

b) Determin a basis for A's column space.

I do this using the spanning set theorem.

Since a3 = a2, a4 = 2a2 + a1 then

B(ColA) = sp{a1,a2, a5} According to the theorem.

c) Next the basis for the B(Null A).

First I row reduce A and then up and write up the set of solutions for A which results in the set B(Null A) = span{(0,-1,1,0) , (1,-2,0,1)}

Is that the correct approach ??

d) There is a vector x = a1 + a2 + a3 + a4 + a5. I'm tasked with showing that this vector belongs to Col A. Finally I'm tasked with finding the vector x with respect to the basis B.

I need some assistance is solving c) and d) therefore I hope there is somebody out there who can guide me :-)

Sincerely and Best Regards,

Fred



/Fred
 
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