Linear algebra: Prove the statement

In summary, the given 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," can be proven by considering the null space and row space of a linear transformation, as well as the dimensions of the given subspaces. This can be done by using the formula \dim U = \dim \ker L + \dim C(L^T) and substituting in the dimensions of the given subspaces.
  • #1
gruba
206
1

Homework Statement


Prove that [itex]\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)[/itex] for every subspace [itex]\mathbb{F}[/itex] and every linear transformation [itex]L[/itex] of a vector space [itex]V[/itex] of a finite dimension.

Homework Equations


-Fundamental subspaces
-Vector spaces

The Attempt at a Solution



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

How to use this theorem to prove the given statement?
 
Physics news on Phys.org
  • #2
You may consider [itex]U=\mathbb{F}+\ker L[/itex] and ##L: U → V##. Then according to your given formula [itex]\dim U = \dim \ker L + \dim C(L^T)[/itex]. 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})##.
 

Related to Linear algebra: Prove the statement

1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and linear transformations to solve various problems in fields such as physics, engineering, and economics.

2. What does it mean to "prove a statement" in linear algebra?

Proving a statement in linear algebra means providing a logical and mathematical explanation or demonstration to show that a certain statement or theorem is true. This involves using axioms, definitions, and previously proven theorems to establish the validity of the statement.

3. How do you approach proving a statement in linear algebra?

The approach to proving a statement in linear algebra involves breaking down the given statement into smaller, more manageable parts. This is followed by using known properties, definitions, and theorems to build a logical argument that supports the statement. It is also important to carefully justify each step and provide clear explanations.

4. What are some common techniques used to prove statements in linear algebra?

Some common techniques used to prove statements in linear algebra include direct proof, proof by contradiction, proof by induction, and proof by construction. These techniques involve using logical reasoning, algebraic manipulations, and definitions to establish the validity of the statement.

5. Are there any tips for successfully proving statements in linear algebra?

To successfully prove statements in linear algebra, it is important to have a strong understanding of the basic concepts and properties of linear algebra. It is also helpful to break down the statement into smaller parts and use clear and concise explanations for each step in the proof. Additionally, practicing regularly and seeking help from resources such as textbooks and online tutorials can also aid in improving proof-writing skills.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
982
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
0
Views
506
  • Calculus and Beyond Homework Help
Replies
1
Views
719
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
524
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Back
Top