Linear Operator Example

Question

Show that the linear operator defined by the equations \begin{align*} w_1 &= x_1 - 2x_2 +x_3\\ w_2 &= 5x_1 - x_2 + 3x_3\\ w_3 &= 4x_1 + x_2 +2x_3\\ \end{align*} is not onto $\mathbb{R}^3$. Find a the relationship between $w_1$, $w_2$, and $w_3$ to ensure the system is consistent and use it to find two vectors in the codomain that are in the range, and two that are not.

Okay, so I got everything except the last half. For the last part of the question, I am not exactly sure what to do but I think I get what they are asking: the "relationship" and how I will use that to find three vectors they ask for.

Anyway, I have proved that the columns of the standard matrix do not span $\mathbb{R}^3$ and therefore the linear transformation does not map onto $\mathbb{R}^3$.

Answer 1

Guide:

Focusing on the RHS, notice that the sum of the first and third expressiongive you the second expressions.

Hence we required $w_1+w_3=w_2$. Try to prove that this condition is what we need.

Answer 2

$$rk\left( \begin{matrix} 1 & -2 & 1 \\ 5 & -1 & 3 \\ 4 & 1 & 2 \end{matrix} \right) \stackrel{w_2 \rightarrow w_2-w_3}{=} rk\left( \begin{matrix} 1 & -2 & 1 \\ 1 & -2 & 1 \\ 4 & 1 & 2 \end{matrix} \right)\stackrel{w_1 \rightarrow w_1-w_2}{=} rk\left( \begin{matrix} 0 & 0 & 0 \\ 1 & -2 & 1 \\ 4 & 1 & 2 \end{matrix} \right) = 2$$