Describe how to find a basis $\beta$ for $R^n$ and a basis $C$ for $R^m$

Question

Let $T : R^n → R^m$ be a linear transformation. Describe how to find a basis $\beta$

for $R^n$ and a basis $C$ for $R^m$ such that matrix for $T$ relative to $\beta$ and $C$ is an

diagonal matrix $P$ of size $m × n$.

Answer 1

This is the well known canonical form of a matrix.

Take $\{\beta_{k+1}, \dots, \beta_n\}$ (where $0\le k \le n$) as a basis of the kernel of $T$. Then according to the rank-nullity theorem, the dimension of the image of $T$ is $k$. Take $\{\beta_1, \dots, \beta_{k}\}$ such that $$\{C_1=T(\beta_1), \dots, C_{k} = T(\beta_{k})\}$$ is a basis of the image of $T$. Complete $\{C_1, \dots , C_{k}\}$ into a basis $\{C_1, \dots C_m\}$ of $\mathbb R^m$.

The two basis are the one you're looking for.