Vectors Generating R^3 Linear Algebra Problem

Determine if a set of vectors generates R^3 by checking for linear independence or solving a system of linear equations.

This linear algebra problem involves determining the basis and dimension of a 2x2 matrix space and a subspace defined by linear equations. The solution involves finding the inverse of a matrix and verifying subspace properties.

Solve a matrix problem involving motor production and component requirements. The solution involves setting up production and component matrices and performing matrix multiplication to find the total components needed.

Solve matrix equations to find the unknown matrix X using matrix inverse and transpose properties.

Find the solution to a system of differential equations given its general solution and an initial condition. The solution involves finding the constants and then substituting them back into the general solution to find the specific solution for x1(t), x2(t), and x3(t).

Find the general solution to a system of differential equations X' = AX, given the eigenvalues and eigenvectors of the matrix A. The solution involves finding linearly independent solutions based on the eigenvalues and eigenvectors, including complex conjugate pairs.

Solve the initial value problem (IVP) for a system of linear differential equations X' = AX, finding the particular solution given the matrix A and initial condition X(0). The solution involves finding eigenvalues, eigenvectors, and applying the initial condition.

The provided text details the process of performing LU decomposition on a 4x4 matrix, including row swaps and elementary row operations to achieve an upper triangular matrix (U) and a lower triangular matrix (L). It also calculates the corresponding diagonal matrix (D) and permutation matrix (P) to complete the decomposition. The solution demonstrates the Gaussian elimination method to find the inverse of a 3x3 matrix.

The eigenvalues of the matrix [[3, 1], [1, 3]] are 4 and 2.

The eigenvalues of the matrix [[0, 1], [-1, 0]] are i and -i.