### Construct a matrix whose column space contains (1, 1, 5) and (0, 3, 1) while it’s null space contains (1, 1, 2).

This question aims to understand the construction of a matrix under given constraints. To solve this question, we need to have a clear understanding of the terms column space and null space. The space which is spanned by the column vectors of a given matrix is called its column space. The space which is spanned […]

### Describe all solutions of Ax=0 in parametric vector form, where A is row equivalent to the given matrix.

This problem aims to get us familiar with vector solutions. To better understand this problem, you should know about the homogeneous equations, parametric forms, and the span of vectors. We can define parametric form such that in a homogeneous equation there are $m$ free variables, then the solution set can be represented as the span of […]

### Determine if b is a linear combination of the vectors formed from the columns of the matrix A.

[ A=begin{bmatrix} 1&-4&2 \ 0&3&5 \ -2&8&-4 end{bmatrix},space b = begin{bmatrix} 3 \ -7 \ -3 end{bmatrix}] This problem aims to familiarize us with vector equations, linear combinations of a vector, and echelon form. The concepts required to solve this problem are related to basic matrices, which include linear combinations, augmented vectors, and row-reduced forms. […]

### Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.

[ boldsymbol{ left [ begin{array}{ c c c } 2 & 5 & 5 \ 5 & 2 & 5 \ 5 & 5 & 2 end{array} right ] ; lambda = 12 } ] The aim of this question is to understand the diagonalization process of a given matrix at given eigenvalues. To solve […]

### For the matrix A below, find a nonzero vector in nul A and a nonzero vector in col A.

[ A = begin{bmatrix} 1 & -2 & 5 & 6 \ 5 & 1 & -10 & 15 \ 1 & -2 & 8 & 4 end{bmatrix} ] This question aims to find the null space which represents the set of all solutions to the homogeneous equation and column space which represents the range of a […]