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 $3x - 5y = -22$.

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 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.

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

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

Determine if a given matrix is an elementary matrix by performing one row operation on an identity matrix to see if it transforms into the given matrix.

Check if a set of vectors in $\mathbb{R}^3$ consisting of (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for $\mathbb{R}^3$.

Given a transformation that projects the space onto the z-axis, where did the vectors in $\mathbb{R}^3$ go, and are there any vectors that were mapped to the zero vector?

Given matrix A which is [3, 1, 4] and matrix B which is [4, 2; 6, 3; 5, 8], multiply matrix A by matrix B, and determine the size and elements of the resultant matrix.

Given matrix A which is $\begin{bmatrix}3 & 4 \\ 7 & 2 \\ 5 & 9\end{bmatrix}$ and matrix B which is $\begin{bmatrix}3 & 1 & 5 \\ 6 & 9 & 7\end{bmatrix}$ , can matrix A be multiplied by matrix B? If so, find the size and elements of the resultant matrix.

Multiply the 2x3 matrix A with the 3x2 matrix B using the row-column rule to obtain the 2x2 matrix AB.

Find the minor for the 3rd row and 2nd column in a given 3x3 matrix A.

Let B be a set of vectors $v_1, v_2, \ldots, v_k$ such that all vectors in B have length 1 $(\|\|v_i\|\| = 1)$ for all $i$ and are orthogonal to each other $(v_i \cdot v_j = 0)$ for $i \ne j$. Show that B is an orthonormal set and prove that B is also linearly independent.