Linear Algebra: Vector Operations and Linear Combinations

Given vectors $V_1 = \begin{bmatrix} 1 \\ -2 \\ -5 \end{bmatrix}$, $V_2 = \begin{bmatrix} 2 \\ 5 \\ 6 \end{bmatrix}$, and $B = \begin{bmatrix} 7 \\ 4 \\ -3 \end{bmatrix}$, determine if there exist scalars $x_1$ and $x_2$ such that $x_1 V_1 + x_2 V_2 = B$.

Describe a plane in $\mathbb{R}^3$ given by the equation $x - 3y + 4z = 0$ in parametric vector form.

Given two vectors, $\mathbf{v} = 2\mathbf{i} - 5\mathbf{j}$ and $\mathbf{w} = -3\mathbf{i} + 7\mathbf{j}$, perform the following operations:

(A) $\mathbf{v} + \mathbf{w}$

(B) $\mathbf{v} - \mathbf{w}$

(C) $2\mathbf{v} + 3\mathbf{w}$

(D) $4\mathbf{v} - 5\mathbf{w}$

Given two vectors $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$, compute $X + Y$.

Subtract the vector $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$ from $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$. Compute $X - Y$.

Calculate the linear combination $3X + 2Y$ for the vectors $X = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $Y = \begin{bmatrix} -4 \\ 6 \end{bmatrix}$.

Given vectors $A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}$, compute $A + B$.

For vectors $A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}$, compute $B - A$.

Calculate the expression $B + 3A$ for vectors $A = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}$.