Question

Consider a 3 x 3 matrix A and a vector $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{V}}$in ${\mathbf{ℝ}}^{\mathbf{3}}$such that${\mathit{A}}^{\mathbf{3}}\mathit{v}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{0}\mathbf{,}$but${\mathit{A}}^{\mathbf{2}}\mathit{v}\mathbf{}\mathbf{⃗}\mathbf{\ne }\mathbf{0}$.

1. Show that the vectors${\mathit{A}}^{\mathbf{2}}\mathit{v}\mathbf{}\mathbf{⃗}\mathbf{,}\mathit{A}\mathit{v}\mathbf{}\mathbf{⃗}\mathbf{,}\mathit{v}\mathbf{}\mathbf{⃗}$forms a basis of ${\mathbf{ℝ}}^{\mathbf{3}}\mathbf{.}$.
2. Find the matrix of the transformationwith respect to the basis.

Short answer

If we have a 3 x 3 matrix A and a vector in such that, but. Then

1. The vectors forms a basis of .
2. The matrix of the transformationwith respect to the basisis-

Where

is a 3 x 3 matrix.

Step-by-step solution

 Definition of linearly independent vectors

Let V be a vector space over a field K. Let, then if there exist scalars, not all of them 0, such that

Then the vectors are called linearly dependent vectors.

Otherwise, linearly independent vectors.