Find a basis for the space of 2×2 lower triangular matrices.

The main objective of this question is to find the basis space for the lower triangular matrices. This question uses the concept of basis space. A set of vectors B is referred to as a basis for a vector space V if each element of V can be expressed as a linear combination of finite […]

It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace corresponding to lambda. Find h in the matrix A below such that the eigenspace for lambda = 4 is two-dimensional.

[ A=begin{bmatrix} 4&2&3&3 \ 0&2 &h&3 \ 0&0&4&14 \ 0&0&0&2end{bmatrix} ] This problem aims to familiarize us with eigenvalues, eigenspace, and echelon form. The concepts required to solve this problem are related to basic matrices which include eigenvectors, eigenspace, and row reduce forms. Now, eigenvalues are a unique set of scalar numbers that are linked […]

Assume that T is a linear transformation. Find the standard matrix of T.

$T:$ $mathbb{R}^2$ → $mathbb{R}^4$, $T(e_1)$ $= (3,1,3,1)$ $and$ $T (e_2)$ $= (-5,2,0,0),$ $where$ $e_1$ $= (1,0)$ $and$ $e_2$ $= (0,1)$ In this question, we have to find the standard matrix of the linear transformation $T$. First, we should recall our concept of the standard matrix. The standard matrix has columns that are the images of […]

Determine if the columns of the matrix form a linearly independent set. Justify each answer.

(begin{bmatrix}1&4&-3&0\-2&-7&4&1\-4&-5&7&5end{bmatrix}) The main objective of this question is to determine whether the columns of the given matrix form a linearly independent or dependent set. If the non-trivial linear combination of vectors equals zero, then the set of vectors is said to be linearly dependent. The vectors are said to be linearly independent if there is […]