Interpreting U={A|A^2=A, A is in M22}: Not a Subspace of M22

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Discussion Overview

The discussion revolves around the interpretation of the set U={A|A^2=A, A is an element of M22} and the assertion that it is not a subspace of M22. Participants are exploring the implications of the condition A^2=A in the context of matrix subspaces, particularly in relation to linear algebra concepts.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • Some participants express confusion about the meaning of A^2=A and its implications for matrix subspaces.
  • One participant notes that the properties required for a subspace apply to matrices similarly to vectors, specifically mentioning closure under addition and scalar multiplication.
  • Questions are raised about whether the sum of two matrices A and B that satisfy A^2=A and B^2=B also satisfies (A+B)^2=A+B.
  • Another question posed is whether scaling a matrix A that satisfies A^2=A by a scalar c results in (cA)^2=cA.

Areas of Agreement / Disagreement

Participants generally agree on the need to understand the properties of subspaces in relation to matrices, but there is no consensus on the interpretation of the specific condition A^2=A or its implications for U being a subspace.

Contextual Notes

The discussion highlights the need for clarity on the definitions and properties of subspaces in the context of matrices, particularly regarding the operations of addition and scalar multiplication.

bbelson01
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How do I interpret the following:

U={A|A^2=A, A is an element of M22} is not a subspace of M22.

I don't quite understand what it's asking in terms of A^2=A. Thanks.
 
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I have moved your new question to a new thread. We ask that new questions are not tagged onto existing threads, as this can make a thread convoluted and hard to follow. You are also more likely to get help in a more timely manner when posting new questions in new threads. :D
 
More vector subspaces

How do I interpret the following: U={A|A^2=A, A is an element of M22} is not a subspace of M22. I don't quite understand what it's asking in terms of A^2=A. Thanks.

That is all the information in the question. I can't make the jump from vectors to matrices in terms of proving subspaces.

Cheers
 
bbelson01 said:
How do I interpret the following: U={A|A^2=A, A is an element of M22} is not a subspace of M22. I don't quite understand what it's asking in terms of A^2=A. Thanks.

That is all the information in the question. I can't make the jump from vectors to matrices in terms of proving subspaces.

Cheers

I have merged this duplicate posting of the question (which was posted in our high school algebra forum) with the pre-existing thread. Vectors and subspaces are university topics, most likely part of a course in linear algebra. Thus, this question is better suited here. :D
 
bbelson01 said:
How do I interpret the following: U={A|A^2=A, A is an element of M22} is not a subspace of M22. I don't quite understand what it's asking in terms of A^2=A. Thanks.

That is all the information in the question. I can't make the jump from vectors to matrices in terms of proving subspaces.
The properties that have to be satisfied by a subspace are the same for spaces of matrices as they are for spaces of vectors. Namely, they must be closed under the operations of addition and scalar multiplication.

In this case, the questions that you need to consider (for $2\times2$ matrices $A$ and $B$) are:
(1) If $A^2 = A$ and $B^2 = B$, is it true that $(A+B)^2 = A+B$?
(2) If $A^2 = A$ and $c$ is a scalar, is $(cA)^2 = cA$?
 

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