Determine if $v=(4,4,-3)$ is a linear combination of the vectors $u_1=(3,1,-1)$ and $u_2=(2,-2,1)$

Question

Determine if $v=(4,4,-3)$ is a linear combination of the vectors $u_1=(3,1,-1)$ and $u_2=(2,-2,1)$.

Answer 1

Hint: Form the matrix $A = \begin{bmatrix} u_1 & u_2 & v \end{bmatrix}$ and row reduce. If you end up with the $3 \times 3$ identity matrix, the three vectors are linearly independent, and there exists no such linear combination. If not, you can read the coefficients of the linear combination off the third column of the row-reduced matrix.

Answer 2

Calculate

$$ \begin{vmatrix} 3&1&-1 \\ 2&-2&1 \\ 4&4&-3 \\ \end{vmatrix}.$$

If this determinant is $0$, then the last row is a linear combination of the first two rows. Otherwise it's not.