Linear algebra: Prove the statement

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1. Feb 22, 2016

gruba

1. The problem statement, all variables and given/known data
Prove that $\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)$ for every subspace $\mathbb{F}$ and every linear transformation $L$ of a vector space $V$ of a finite dimension.

2. Relevant equations
-Fundamental subspaces
-Vector spaces

3. The attempt at a solution

Theorem:
If $L:U\rightarrow V$ is a linear transformation and $\dim U=n$, then $\dim Ker L+\dim C(L^T)=n$. $Ker L$ is the null space, $C(L^T)$ is the row space of $L$ and $n$ is the number of column vectors in $[L]$.

How to use this theorem to prove the given statement?

2. Feb 22, 2016

Staff: Mentor

You may consider $U=\mathbb{F}+\ker L$ and $L: U → V$. Then according to your given formula $\dim U = \dim \ker L + \dim C(L^T)$. With $\dim C(L^T) = \dim L(U) = \dim L(\mathbb{F} + \ker L) = \dim L(\mathbb{F})$ we have $\dim(\mathbb{F}+\ker L) = \dim U = \dim \ker L + \dim L(\mathbb{F})$.

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