Applied Linear Algebra | Linear Dependent Matrix

Question

Question:

My response:

Am I solving the above question correctly? Or am I on the wrong path? Thank you for your help.

Answer 1

Your method works fine.

Here's another suggestion, which you may or may not find easier. We are looking for numbers $\lambda_1$, $\lambda_2$, such that

$$\lambda_1\begin{bmatrix}1 \\ 3 \\ -1 \end{bmatrix}+\lambda_2\begin{bmatrix}-5 \\ -8 \\ 2 \end{bmatrix} = \begin{bmatrix}3 \\ -5 \\ k \end{bmatrix}.$$

The first two rows of this vector equation gives two linear equations in two unknowns $\lambda_1$ and $\lambda_2$.

Thus you can solve the two equations:

$$\lambda_1 - 5\lambda_2 = 3,$$ $$3\lambda_1 - 8\lambda_2 = -5.$$

With standard methods, you get $\lambda_1 = -7$ and $\lambda_2 = -2$. From the third row of the vector equation above, you can plug in these values for $\lambda_1$ and $\lambda_2$ to obtain $k$:

$$k = \lambda_1\cdot (-1) + \lambda_2\cdot 2 = -7\cdot(-1) + (-2)\cdot 2 = 7 - 4 = 3.$$