Linear Algebra

Solve the system of equations: $7x + 5y = -12$ and $3x - 4y = 1$ using any method such as graphing, elimination, and substitution.

Graph the two parallel lines representing the linear equations and determine if the system has no solution.

Using Gauss-Jordan elimination, solve the system of linear equations given by the augmented matrix: $\begin{bmatrix} 1 & -1 & 3 & | & 8 \\ 1 & 1 & 2 & | & 1 \\ 3 & 1 & 4 & | & 10 \end{bmatrix}$

Given the vector $v = (2, 3)$ in the standard basis, find the new coordinates $v'$ in the basis defined by $u_1 = (1, 2)$ and $u_2 = (3, 3)$.

Consider the bases for $\mathbb{R}^2$: $B = \{ [1, 1], [0, 2] \}$ and $C = \{ [0, 1], [1, 2] \}$. If the coordinate vector for $x$ with respect to $B$ is $[3, 1]$, find the coordinate vector for $x$ with respect to $C$.

Determine if the set of vectors $(1, -2, 0)$, $(4, 0, 8)$, $(3, -1, 5)$ is linearly independent or dependent by performing row reduction.

Solve the following system of equations using Cramer's Rule: $-x_1 + 4x_2 + 3x_3 = 2$, $2x_2 + 2x_3 = 1$, $x_1 - 3x_2 + 5x_3 = 0$.

Using matrices and Cramer's Rule, solve for the values of $x$, $y$, and $z$ given the system of equations: $3x + 3y + 5z = 1$, $5x + 9y + 17z = 0$, $3x + 9y + 5z = 0$.

Determine if a given matrix is diagonalizable and, if so, diagonalize it.

Raise the matrix $A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \end{bmatrix}$ to the 100th power using diagonalization.

Diagonalize the matrix A where A is given as $\begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{bmatrix}$.

Consider the matrix $A$, with entries $-3, -4; 5, 6$. We must first find the eigenvalues, which means we must solve for the values of $\lambda$ that satisfy this expression, where the determinant of $A - \lambda I = 0$.

Diagonalize the following 3x3 matrix.

For a given matrix $A$, find the eigenvalues $\lambda$ and corresponding eigenvectors $X$ such that $A X = \lambda X$.

Find the eigenvalues and eigenvectors of the given matrix.

Determine if the given matrix is an Elementary matrix by checking if it can be obtained from the identity matrix using a single Elementary row operation.

Determine if the given set of four matrices, with specific ones and zeros, span $R^{2x2}$ and form a basis.

$\begin{bmatrix}1 & 1 \\0 & 0\end{bmatrix}$ $\begin{bmatrix}0 & 0 \\1 & 1\end{bmatrix}$ $\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}$ $\begin{bmatrix}0 & 1 \\1 & 1\end{bmatrix}$

Check comprehension related to vector spaces, subspaces, span, linear independence, basis, and dimension.

Assume we have five websites numbered 1, 2, 3, 4, and 5. Determine which website will have more traffic using the PageRank algorithm. Construct the Google Matrix for these websites and analyze it to find the relative importance or ranking of each website based on their traffic. Use eigenvalues and eigenvectors for the analysis.

Using a 2x2 matrix, identify eigenvectors and eigenvalues. Explain the transformation effects on vectors when multiplied with this matrix, including stretching, shrinking, and rotation dynamics. Consider cases with real and non-real eigenvalues and describe their implications for vector transformations.