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

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SUMMARY

The set U={A|A^2=A, A is an element of M22} is definitively not a subspace of M22, as it fails to satisfy the closure properties required for subspaces in linear algebra. Specifically, for two matrices A and B in U, the sum (A+B) does not necessarily satisfy (A+B)^2 = A+B, and for a scalar c, (cA)^2 does not equal cA. These properties must hold for any subset to qualify as a subspace, which U does not meet.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly vector spaces and subspaces.
  • Familiarity with matrix operations, specifically for 2x2 matrices.
  • Knowledge of closure properties under addition and scalar multiplication.
  • Ability to interpret matrix equations, particularly idempotent matrices (where A^2 = A).
NEXT STEPS
  • Study the properties of vector spaces and subspaces in linear algebra.
  • Learn about idempotent matrices and their implications in matrix theory.
  • Explore closure properties in the context of matrix addition and scalar multiplication.
  • Investigate examples of subspaces in M22 to reinforce understanding of the criteria for subspaces.
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Students and educators in linear algebra, mathematicians exploring matrix theory, and anyone seeking to understand the properties of subspaces in the context of matrices.

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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