Answer
See explanation
Work Step by Step
Define vectors as coefficient vectors in front of every unknown.
$v_{1}=\left[\begin{array}{c}2 \\ -5 \\ 7\end{array}\right], v_{2}=\left[\begin{array}{c}-4 \\ 1 \\ -5\end{array}\right], v_{3}=\left[\begin{array}{c}-2 \\ 1 \\ -3\end{array}\right], b=\left[\begin{array}{l}b_{1} \\ b_{2} \\ b_{3}\end{array}\right]$
Now determine if vectors $v_{1}, v_{2}, v_{3}$ span $\mathbb{R}^{3}$
Find reduced echelon form of: : : : : reduced echelon form o form e: :
$\left[\begin{array}{ccc}2 & -4 & -2 \\ -5 & 1 & 1 \\ 7 & -5 & -3\end{array}\right],$ multiply first row with $5 / 2,-7 / 2$ and add to second and
$\sim\left[\begin{array}{ccc}2 & -4 & -2 \\ 0 & -9 & -4 \\ 0 & 9 & 4\end{array}\right],$ add second row to third
$\sim\left[\begin{array}{ccc}2 & -4 & -2 \\ 0 & -9 & -4 \\ 0 & 0 & 0\end{array}\right]$
since there is no pivot in third row, vectors $v_{1}, v_{2}, v_{3}$ do not span $\mathbb{R}^{3}$.
b)
Define coefficient matrix $A=\left[\begin{array}{ccc}2 & -4 & -2 \\ -5 & 1 & 1 \\ 7 & -5 & -3\end{array}\right]$ and right side vector $b=\left[\begin{array}{l}b_{1} \\ b_{2} \\ b_{3}\end{array}\right]$
Now determine if $b$ is a linear combination of columns of $A$, or if columns of $A$ span $\mathbb{R}^{3} .$ You can do it by same way as in part a).
C)
Define $T(x)=A x, \text { with } A \text { as in part } b)$
Now determine if range of $A$ is $\mathbb{R}^{3}$.
