How Can You Represent a Linear Transformation with a Block Matrix Form?

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Homework Help Overview

The discussion revolves around representing a linear transformation using a block matrix form. The original poster presents a problem involving dimensions of vector spaces and the existence of specific bases for these spaces.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the dimensions provided in the problem statement and question the relevance of certain terms. There is an attempt to clarify the correct dimensions of the vector spaces involved.

Discussion Status

The discussion is currently focused on clarifying the problem statement and ensuring that the dimensions of the vector spaces are correctly identified. Some participants are seeking hints and guided questions to approach the problem.

Contextual Notes

There is a noted confusion regarding the dimensions of the vector spaces, specifically the initial misstatement of "dim(A)=n" instead of "dim(C)=n". This has led to questions about the purpose of the statements in the problem.

tiger2030
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Homework Statement


Let T: C→D with dim(A)=n and dim(B)=m. Show that there exists bases B and B' for C and D, respectively, such that the matrix of T in block form is

M=|I 0|
|0 0|

where I is a k by k identity matrix

Homework Equations


The Attempt at a Solution


Honestly no idea where or how to start. Just looking for some hints and guided questions
 
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What is the purpose of the "dim(A)=n" statement.
A is never used.
Could you check the problem statement?
 
Sorry for the mix up. It should be dim(C)=n and dim(D)=m
 
So there exists {u1,u2,...,un} such that for any c that's an element of C we can make a linear combination of {u1,...un} that equals c. Similarly for {v1,v2,...vm} for any d that's an element of D
 

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