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

In summary: The first question is easy to answer; the result is always true.The second question is a bit more complicated. If $A$ and $B$ are two 2-by-2 matrices, then $(A+B)^2 = A+B$ because the two matrices are added together. However, if $A$ and $B$ are any two matrices, then there are a lot of cases where the result won't be $A+B$. For example, if $A$ is a 3-by-3 matrix and $B$ is a 2-by-2 matrix, then $(A+B)^2 = A+B$ because the two matrices are added together
  • #1
bbelson01
6
0
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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  • #2
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
 
  • #3
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
 
  • #4
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
 
  • #5
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 $(cA)^2 = cA$?
 

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

1. What does the notation "U={A|A^2=A, A is in M22}" mean?

The notation "U={A|A^2=A, A is in M22}" represents a set of matrices, denoted by U, where each matrix A in the set satisfies the condition A^2=A and is in the set of all 2x2 matrices, denoted by M22.

2. Why is U not considered a subspace of M22?

U is not considered a subspace of M22 because it does not satisfy all three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. In this case, U does not contain the zero matrix, which is a requirement for a subspace.

3. Can you provide an example of a matrix in U?

One example of a matrix in U is the identity matrix, I, since I^2=I and I is a 2x2 matrix.

4. How can you determine if a set is a subspace of another set?

To determine if a set is a subspace of another set, you can check if it satisfies all three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector. If any of these properties are not satisfied, then the set is not a subspace.

5. What is the significance of U not being a subspace of M22?

The fact that U is not a subspace of M22 means that it does not follow the same rules and properties as a subspace. This can have implications in mathematical calculations and proofs, as well as in practical applications. It also highlights the importance of carefully defining and understanding mathematical concepts and notations.

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