Question

determine if the vector b is in teh span of the columns of the matrix A
A= 1 2 3
4 5 6
7 8 9
b= 10
11
12

Answer 1

I *believe* the way to do this is to see if there is a multiple of each of the column vectors of \(\bf{A}\) that will produce \(\vec{b}\).

Answer 2

But don't quote me on that!

Answer 3

do you row reduce it?

Answer 4

Hm, no a multiple of the sums. So I believe you have to break down the columns into three vectors:

\[\begin{align}
\bf{A} = \left[\begin{matrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{matrix}\right]
\end{align}\]

You turn this into three vectors:

\[\begin{align}
\vec{a_1} = \left[\begin{matrix}1\\4\\7\end{matrix}\right] &
\vec{a_2} = \left[\begin{matrix}2\\5\\8\end{matrix}\right] &
\vec{a_3} = \left[\begin{matrix}3\\6\\9\end{matrix}\right]
\end{align}\]

Answer 5

Then, if you can find an \(x, y, z\) such that:

\[x\vec{a_1} + y\vec{a_2} + z\vec{a_3} = \vec{b}\]

Then I believe \(\vec{b}\) is in the span of the columns of \(\bf{A}\).

Answer 6

You end up with a system of 3 equations, one for each of the rows multiplied by x/y/z, and you can solve to see if there is a solution for x/y/z.

Answer 7

then do u do the agumented of row reduced?

Answer 8

I have no idea! Hehe. Like I said, I don't remember *too* much about this, and I've posted more or less everything I do remember. Sorry :/

Answer 9

okay thank you :)