Given the vectors $\alpha_1=(1,0,-1)$, $\alpha_2=(1,2,1)$, $\alpha_3=(0,-3,2)$, express each of the standard basis vectors as linear combination of $\alpha_1$, $\alpha_2$, and $\alpha_3$.
[Math] Express each of the standard basis vectors as linear combination of $\alpha_1,\alpha_2,\alpha_3$
linear algebravector-spaces
Best Answer
Hint: Construct a matrix with those vectors as columns, and try to convert your question into the solution of a matrix equation.
Letting the $\alpha_i$ be written as column vectors, use the fact that $$\lambda_1\alpha_1+\lambda_2\alpha_2+\lambda_3\alpha_3=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & -3 \\ -1 & 1 & 2 \end{pmatrix}\begin{pmatrix}\lambda_1\\ \lambda_2 \\ \lambda_3 \end{pmatrix}.$$