Linear algebra: Prove the statement

[gruba]

Homework Statement

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.

Homework Equations

-Fundamental subspaces
-Vector spaces

The Attempt at a Solution

Theorem: [/B]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?

 

[fresh_42]

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 $\dim(\mathbb{F}+\ker L) = \dim U = \dim \ker L + \dim L(\mathbb{F})$.