Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r

In summary: LaTeX command. ^ is used for exponents.In summary, Axler discusses the concept of a matrix of a linear map and matrix multiplication in his book "Linear Algebra Done Right." He then introduces Theorem 3.43, which states that for linear maps T and S, the matrix of their product ST is equal to the product of their individual matrices M(S) and M(T). He proves this theorem by considering the bases of the vector spaces U, V, and W and using the definition of matrix multiplication.
  • #1
MidgetDwarf
1,488
626
∈Was wondering if anyone here could help me with an explanation as to how Axler arrived at a particular step in a proof.

These are the relevant definitions listed in the book:

Definition of Matrix of a Linear Map, M(T):

Suppose ##T∈L(V,W)## and ##v_1,...,v_n## is a basis of V and ##w_1 ,...,w_m## is a basis of W. The matrix of T with respect to these bases is the m-by-n matrix M(T) whose entries ##A_j , _k## are defined by ## T_v_k = A_1,k w_1 + ... +A_m,k w_m ##

Definition of Matrix Multiplication:

Suppose A is an m-by-n matrix and C is an n-by-p matrix. Then AC is defined to the m - by- p matrix whose entry in row j, column k, is given by the following equation: ## (AC)_{j,k} = \sum_{r=1}^n A_j,r C_r,k ##

Now for the Theorem of the proof I need help with.

Theorem 3.43 (page 74-75): The Matrix Of The Product Of Linear Maps:

If T∈L(U,V) and SεL(V,W) , then M(ST)=M(S)M(T).

Proof:

Assume ## v_1 , ... , v_n ## is a basis of V and ##w_1 , ... , w_m ## is a basis of W. Suppose also that we have another vector space U and that ## u_1 ,..., u_p ## is a basis of U. Consider linear maps T : U →V and S : V→W. ( I proved easier that the composition of linear maps is a linear maps)

Suppose M(S) = A and M(T) + C. For 1≤ k ≤ p , we have

##(ST)u_k ## = ## S(\sum_{r=1}^n C_r,k v_r ) = \sum_{r=1}^n C_r,k Sv_r ## ##= \sum_{r=1}^n C_r,k \sum_{j=1}^m A_j,r w_j##
 
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  • #2
Hi. There seem to be some formatting issues with the post. Please resolve those.
 
  • #3
Wow my Latek came out wrong. Ill try some practice with easier code, before reposting this question. If anyone is interested, it is the 3rd line to 4th line on page 74 of Axler Linear Algebra Done Right 3rd ed.
 
  • #4
MidgetDwarf said:
Wow my Latek came out wrong. Ill try some practice with easier code, before reposting this question. If anyone is interested, it is the 3rd line to 4th line on page 74 of Axler Linear Algebra Done Right 3rd ed.
Please rewrite your question in a new thread. Not all of us have the book.

I have tried to correct your post, but you made too many mistakes and I didn't always know what was meant. Maybe you should read
https://www.physicsforums.com/help/latexhelp/
again and make more use of the preview function.
 
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  • #5
Math_QED said:
There seem to be some formatting issues with the post. Please resolve those.
I have fixed some of the problems.
The main problems I saw were multiple subscripts and using \n for exponents.
 

Related to Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r

1. What is the purpose of a line of proof in Axler Linear Algebra Done Right 3rd edition?

The purpose of a line of proof is to provide a logical and step-by-step explanation of a mathematical concept or theorem. It allows the reader to follow the thought process and understand how the conclusion was reached.

2. How do I know if a line of proof is valid?

A valid line of proof follows the rules of logic and uses mathematical principles and definitions accurately. It should also be clear and easy to follow, with each step logically leading to the next.

3. Can a line of proof have multiple correct solutions?

No, a line of proof should have only one correct solution. However, there may be multiple ways to arrive at the same solution, and different authors may use different approaches or notation to explain the same concept.

4. What should I do if I don't understand a line of proof?

If you do not understand a line of proof, you can try breaking it down into smaller steps or looking up any unfamiliar terms or symbols. It may also be helpful to consult other resources or ask for clarification from a teacher or tutor.

5. Are there any common mistakes to watch out for when writing a line of proof?

Some common mistakes to watch out for when writing a line of proof include incorrect use of symbols or notation, skipping steps, and making assumptions without proper justification. It is important to be thorough and precise when writing a line of proof to avoid these errors.

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