Linear Algebra

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.

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

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.

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

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.

Solve the system of linear equations using the Gauss-Jordan elimination method.

Using Gaussian elimination, solve the system of linear equations represented by the matrix.

Explain the concept of Hilbert space in the context of quantum mechanics.

Using the least squares method, solve for the best-fit line given a set of data points.

Given a matrix, perform row reduction to determine the rank and nullity, ensuring the rank plus nullity equals the number of columns in the matrix.

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.