(adsbygoogle = window.adsbygoogle || []).push({}); So the theorem says:

Suppose that ##U## and ##V## are finite dimensional vector spaces, and that ##T:U\to V##, ##S: V \to W##. Then

##\text{dim Ker }ST \le \text{dim Ker }S + \text{dim Ker }T##.

Proof:

Set ##U_0 = \text{Ker }ST## and ##V_0 = \text{Ker }S##. ##U_0## and ##V_0## are subspaces of ##U## and ##V##. Since ##ST=0## on ##U_0## we have ##T(U_0) \subset V_0##. We can then consider ##T## and ##S## as mappings defined on ##U_0## respectively ##V_0##. Since ##\text{Ran }T## (Range, column space, image etc.) is a subspace to ##V_0## we have ##\text{dim Ran }T \le \text{dim }V_0 = \text{dim Ker }S##. But according to the Rank-nullity theorem, ##\text{dim Ker }T + \text{dim Ran }T = \text{dim }U_0##.

Hence ##\text{dim Ker }ST = \text{dim }U_0 = \text{dim Ker T} + \text{dim Ran }T \le \text{dim Ker }T + \text{dim Ker }S##.

(I translated this so it's possible worse worded than originally.)

I can't really follow the steps in this proof, no matter how many times I look through it I don't follow all the steps. Starting it with

##U_0## and ##V_0## are subspaces of ##U## and ##V##

The word "and"" confuse me here. It's supposed to mean that ##U_0## to ##U## and ##V_0## to ##V## and nothing else right?

We can then consider ##T## and ##S## as mappings defined on ##U_0## respectively ##V_0##.

What does this mean? That for our purposes we can consider ##T:U_0 \to V## and ##S:V_0 \to W##? Why?

Since ##\text{Ran }T## is a subspace to ##V_0##

I understand this to be true if we indeed consider ##T:U_0 \to V## since by definition ##T(U_0)## maps into ##V_0## but not for ##U## in general.

And then##\text{dim Ker }T + \text{dim Ran }T = \text{dim }U_0##.I don't get how we're allowed to only consider ##U_0## instead of ##U##.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Understanding proof for theorem about dimension of kernel

Loading...

Similar Threads for Understanding proof theorem |
---|

I Understand homomorphisms from Z^a --> Z^b |

I Understanding Hilbert Vector Spaces |

A Is the proof of these results correct? |

I Doubt about proof on self-adjoint operators. |

B Help understanding a proof |

**Physics Forums | Science Articles, Homework Help, Discussion**