Linear Algebra

Given the vectors $v_1 = \left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)$ and $v_2 = \left(\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}\right)$, determine if they form an orthonormal set in $\mathbb{R}^3$.

Multiply the scalar 3 with the matrix $\begin{bmatrix} 7 & 5 & -10 \\ 3 & 8 & 0 \end{bmatrix}$.

Determine whether the following homogeneous system has non-trivial solutions by inspection: 3 equations with 4 unknowns (X1 through X4). Since there are more unknowns than equations, it is guaranteed that this homogeneous linear system will have infinitely many solutions.

Solve the system of equations: $5x + 4y = 0$ and $2x - 2y = 0$, using elimination by addition to find the trivial solution.

Let V be a vector space. Verify whether a subset S, which is made of vectors of the form (x, 0, -x), is a subspace of V by checking the properties of closure.

What is the transpose of matrix A?

What is B transpose going to be equal to?

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 $B = \begin{bmatrix} 0 \\ 2 \\ 7 \end{bmatrix}$.