$A\in M_{m,n}(F)$ we know that $\dim\ker A +\dim \operatorname{ran} A=n$. Suppose $\dim \ker A=r$,and $\{v_{1},...,v_{r}\}$ is a basis for the nullspace of A.It can be extended to a basis $\{v_{1},...,v_{n}\}$ of $F^{n}$. how to prove that $Av_{r+1},...,Av_{n}$ is a basis for the range of A?
how to prove $Av_{r+1},...,Av_{n}$ is a basis for the range of A?
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0What have you tried? It is rather immediate, if you write a generic vector as $v = \sum_{k=1}^n \lambda_k v_k$. Anyway, the complete proof is here: http://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem – 2012-09-17
2 Answers
Since $\{v_1, \ldots , v_n\}$ is a basis, we can write every element in $F^n$ uniquely as:
$ F^n = \{a_1 v_1 + \cdots + a_n v_n: a_1, \ldots, a_n \in F\} \\ $
Also, we have: \begin{align*} \operatorname{ran} A &= \{Av : v \in F^n\} \\ &= \{A(a_1 v_1 + \cdots + a_n v_n) : a_1, \ldots, a_n \in F \} \\ &= \{A(a_1 v_1 + \cdots + a_r v_r + a_{r+1} v_{r+1} + \cdots + a_n v_n) : a_1, \ldots, a_n \in F \} \\ &= \{A(a_1 v_1 + \cdots + a_r v_r) + A(a_{r+1} v_{r+1} + \cdots + a_n v_n) : a_1, \ldots, a_n \in F \} \\ &= \{0 + A(a_{r+1} v_{r+1} + \cdots + a_n v_n) : a_{r+1}, \ldots, a_n \in F \} \\ &= \{A(a_{r+1} v_{r+1} + \cdots + a_n v_n) : a_{r+1}, \ldots, a_n \in F \} \\ &= \{a_{r+1} Av_{r+1} + \cdots + a_n Av_n : a_{r+1}, \ldots, a_n \in F \} \end{align*}
Where: $a_1 v_1 + \cdots + a_r v_r \in \operatorname{ker} A$. Therefore: $A(a_1 v_1 + \cdots + a_r v_r) = 0$.
This shows that $\{Av_{r+1}, \ldots, Av_n\}$ spans $\operatorname{ran}A$. Can you show it's independent?
HINTS: Obviously $\operatorname{span}\{Av_{r+1},\dots,Av_n\}\subseteq\operatorname{ran}A$, so you need to prove two things: that the span is all of the range of $A$, and that the vectors $Av_{r+1},\dots,Av_n$ are linearly independent. The first of these is very easy, since everything in the range of $A$ is of the form $A\sum_{k=1}^n\alpha_kv_k$. For the second, suppose that $\alpha_{r+1}Av_{r+1}+\ldots+\alpha_nAv_n=0$. Then
$\begin{align*} 0&=\alpha_{r+1}Av_{r+1}+\ldots+\alpha_nAv_n\\ &=A(\alpha_{r+1}v_{r_1}+\ldots+\alpha_nv_n)\;, \end{align*}$
what does this tell you about $\alpha_{r+1}v_{r_1}+\ldots+\alpha_nv_n$?