Question

Determine if b is a linear combination of the vectors formed from the columns of the matrix A.
[ 1 0 5] [ 2]
A=[-2 1-6], b=[-1]
[ 0 2 8] [ 6]

Answer 1

\[\left[\begin{matrix}x_1 & x_2 & x_3 \\y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{matrix}\right]\left[\begin{matrix}a\\b\\c\\\end{matrix}\right]=a\left[\begin{matrix}x_1\\y_1 \\ z_1\end{matrix}\right]+b\left[\begin{matrix}x_2\\y_2 \\ z_2\end{matrix}\right]+c\left[\begin{matrix}x_3\\y_3 \\ z_3\end{matrix}\right]\]If there exist some nontrivial \(a, b, c\) for which the rhs of the above is equal to \(b=\left[\begin{matrix}s_1\\s_2\\s_3\end{matrix}\right]\), then the column matrix \(b\) is a linear combination of the columns. To check, take the three linear equations that you get from the rhs of the above.