Question

For what values of h is v3 in Span{v1,v2}

Answer 1

\[v1=\left[\begin{matrix}1 & \\ -3 & \\2 & \end{matrix}\right]\]
\[v2=\left[\begin{matrix}-3 & \\ 9 & \\-6 & \end{matrix}\right]\]
\[v3=\left[\begin{matrix}5 & \\ -7 & \\h & \end{matrix}\right]\]

Answer 2

for v3 to be in span{v1,v2}, it must be expressible as a linear combination of v1 and v2 or in other words:
v3=c1v1+c2v2
where c1 and c2 are constants. so we'll have the matrix:
\[\left[\begin{matrix}1 & -3 \\ -3 & 9 \\ 2 & -6\end{matrix}\right]\left(\begin{matrix}c1 \\ c2\end{matrix}\right)=\left(\begin{matrix}5 \\ -7 \\h\end{matrix}\right)\]
solving this will give you equations
c1-2c3=5
0=8
0=h-10
notice that because of the error 0=8, then even though whatever value of h you put, v3 can never be expressed as a linear combination of v1 and v2, hence it cannot be in the span{v1,v2}. So there are no values of h which will make v3 be in span{v1,v2}