Linear Algebra

Solve the following homogeneous linear system using Gaussian elimination. The system matrix is provided, and the constants are all zero. Determine the type of solutions (trivial or infinitely many) the system possesses.

Given a homogeneous system of linear equations in the form $A \mathbf{x} = 0$, describe the solution set in parametric vector form.

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

Solve the system of equations: $x_1 - 2x_2 + x_3 = 0$, $6x_2 - 3x_3 = 0$, and $x_1 - 2x_2 - x_3 = 0$, using row reduction to demonstrate that it only has the trivial solution.

Put these three vectors into a matrix, row reduce it, and identify how many pivots we get to determine the dimension of the span.

Given a set of four vectors in $\mathbb{R}^2$, put them as columns of a matrix, row reduce, and identify the number of pivots to determine the dimension of the span.

Find the span of these two vectors in $\mathbb{R}^4$ using the row reduction technique to determine the dimension.

Determine the dimension of the span for these three vectors in $\mathbb{R}^3$ by putting them in a matrix and row reducing to find the number of pivots.

Given three vectors from $\mathbb{R}^3$, which are (2, 1, -1), (0, 2, 2), and (-1, -1, -1), determine their span by forming the linear combination $a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3}$ where $a, b,$ and $c$ are scalars.

Is a given vector $W = \begin{bmatrix} 1 \\ 2 \\ 4 \end{bmatrix}$ in the span of two vectors $U = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$ and $V = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$?

Given the set U consisting of vectors from $\mathbb{R}^3$ defined by the equations:

$a = 2r - s, \quad b = 3r, \quad c = r + s$ where $r, s$ are real numbers, determine whether U is a subspace of $\mathbb{R}^3$.

Let $W = \{ [a + 2b, a - b, 3b] \mid a, b \in \mathbb{R} \}$. Determine if $W$ is a subspace of $\mathbb{R}^3$.

Given a 2D integer array, return the transpose of the matrix.