# Column Space and Linear Combinations Question

• June 4th 2009, 07:15 PM
Phatmat
Column Space and Linear Combinations Question
I am stuck on how to work out this question.

1) In each case determine whether b is in the column space of A, and if so, express b as a linear combination of the column vectors A.

http://img13.imageshack.us/img13/5123/47311236.jpg

My answer so far to (b):
http://img189.imageshack.us/img189/9657/banse.jpg

Is someone able to let me know if this column space is correct?

My main question: If so, how am I supposed to find out if b is in the column space?

Thanks.
• June 4th 2009, 07:34 PM
Random Variable
The column space is just the space spanned by the columns of A. (But in this problem they also form a basis for the column space because they're linearly independent.)

If b is in the column space of A, then there exits an $\alpha$ , $\beta$ and $\gamma$ such that

$\alpha \left(\begin{array}{c} 1 \\ 9 \\ 1\end{array}\right) + \beta \left(\begin{array}{c} -1 \\ 3\\ 1\end{array}\right) + \gamma \left(\begin{array}{c} 1 \\ 1 \\ 1\end{array}\right) = \left(\begin{array}{c}-5 \\ 1\\ -1\end{array}\right)$