We have a linear transformation $R^3 \rightarrow R^3$ where in a standard basis there is a matrix:
$$A =\begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 1\\ 2 & 0 & 1 \end{bmatrix}$$
1.)What matrix does belong to this linear transformation in a basis:
$$B = \left\{\quad\begin{bmatrix} 0 \\ 1\\ 2 \end{bmatrix} ,\ \begin{bmatrix} 1 \\ 0\\ -1 \end{bmatrix} ,\ \begin{bmatrix} 1 \\ 1\\ 0 \end{bmatrix}\quad \right\}$$
My understanding of this:
1.The standard basis consists of vectors that are independent of each other. How can then a matrix be in standard basis. Does the matrix A consist of these vectors?
2.What does the $R^3\rightarrow R^3$ have to do with this problem , what if the question would be $R^3\rightarrow R^2$ or $R^2\rightarrow R^3$.
- So a linear transformation is a linear mapping that maps the zero vector to a zero vector. I read all the theory behind it, but still cant figure out the initial problem