Linear Algebra

Calculate the determinant of the 3x3 matrix: $\begin{pmatrix} 5 & 7 & 8 \\ 4 & -3 & 6 \\ 1 & 7 & 9 \end{pmatrix}$.

Consider the following two equations: $2x + 3y = 8$ and $5x - 3y = -1$. How can we solve these two equations using elimination?

Solve the system of equations using elimination: $2x + 5y = 19$ and $x - 2y = -4$.

Given the equations $y = 5 - 2x$ and $4x + 3y = 13$, solve for $x$ and $y$ using substitution.

Use substitution to solve the system of equations: $y = 3x + 2$ and $y = 7x - 6$.

Solve for $x$ and $y$ using substitution in the following system: $4x + 2y = 14$ and $3x - 5y = -22$.

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

Given two linear equations in standard form, solve the system of equations by graphing to find the intersection point.

Given two linear equations in slope-intercept form, solve the system of equations by graphing to find the intersection point.

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

Graph coincident lines of the linear equations to show that the system has infinitely many solutions.

Using the criteria for linear dependence without division, determine if the columns of a given 2x2 matrix are linearly dependent.

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

What if we want to graph $x^2 + 2$?

What if we have the function $-2(x-3)^2 + 4$?

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: $3x + 3y + 5z = 1$, $5x + 9y + 17z = 0$, $3x + 9y + 5z = 0$.