Linear Algebra

For a transpose matrix $A^T$ of a 4x5 matrix $A$, find the rank and nullity of $A^T$ and verify the rank-nullity relation.

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

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.

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.

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 $\mathbb{R}^3$.

What is the transpose of matrix A?

What is B transpose going to be equal to?

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