Is singular matrix is a subspace of vector space V?

In summary: So if det(A)=0, then det(xA)=0. In summary, if a matrix is singular, then it does not belong to S, which is the subset of V that includes the identity matrix.
  • #1
Maxwhale
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Homework Statement



S is a subset of vector space V,

If V is an 2x2 matrix and S={A|A is singular},

a)is S closed under addition?
b) is S closed under scalar multiplication?


Homework Equations


S is a subspace of V if it is closed under addition and scalar multiplication.


The Attempt at a Solution


I tried to use the definition of sinularity. i.e. a matrix in not invertible. But could not decide if it was closed under addition and scalar multiplication.
 
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  • #2
Well, play around with examples of singular matrices. It's not too hard to find two that add up to an invertible one; in fact, you can find two singular matrices that add up to the identity.
 
  • #3
yeah, i can get an identity if i add [1 0; 0 0] and [0 0; 01]. So what does that insinuate? If we add two we get an invertible matrix, which implies that the solution to AX=B is unique. But how do I get to point where i can decide if they are closed under addition and scalar multiplication? Further help will be highly appreciated.

thank you
 
  • #4
What does it mean for S to be closed under addition and scalar multiplication?
 
  • #5
if vec u and vec v are in S, vec u + vec v also should be in S (closed under addition)
for any scalar r, r(vec u) = r*vec(u) and lies in S(closed under scalar multiplication)
 
  • #6
OK, now think about what you did: you added two things in S and got the identity matrix. What does this tell you about S?
 
  • #7
the identity matrix belongs to S. So u and v are closed under addition. Right?
 
  • #8
Really -- does the identity matrix belong to S?
 
  • #9
ohh... so S is a singular, meaning that it is not invertible. The identity matrix is invertible, hence does not belong to S. I hope i got it right this time
 
  • #10
Yup. So S isn't closed under addition.

Now, let's turn to scalar multiplication. Suppose we have a singular matrix A and we multiply it by a scalar r. Can rA ever be nonsingular (i.e. invertible)? Suppose it can be - what does this tell you about A?
 
  • #11
again, if we turn it into an invertible matrix, it would mean that A does not belong to S.
But i did not get an invertible matrix by multiplying a singular matrix with a scalar. Say A = [1 0; 0 0], if we multiply by r, we get, rA= [r 0; 0 0], which is still singular, right?
 
  • #12
More succinctly, if det(A)= 0, then det(xA), for any scalar x, is what?
 
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  • #13
we have not done determinant so far. But i believe in the above case, det(xA) should equal 0. I think so because det(xA) = x* det(A).
 
  • #14
Yes, that's correct.
 

1. Is a singular matrix always a subspace of vector space V?

No, a singular matrix is not always a subspace of vector space V. A subspace must contain the zero vector and be closed under vector addition and scalar multiplication, but a singular matrix may not fulfill these requirements.

2. Can a subspace of vector space V contain only singular matrices?

No, a subspace of vector space V can contain a combination of both singular and non-singular matrices. As long as the subspace fulfills the requirements of a subspace, it can be a mix of both types of matrices.

3. How can we determine if a singular matrix is a subspace of vector space V?

To determine if a singular matrix is a subspace of vector space V, we must check if it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication. If all three conditions are met, then the singular matrix is a subspace of vector space V.

4. What is the significance of a singular matrix being a subspace of vector space V?

If a singular matrix is a subspace of vector space V, it means that the matrix is a subset of the vector space and follows the same rules and properties as the vector space. This can be useful in solving linear systems and understanding the behavior of matrices within a vector space.

5. Can a singular matrix be a proper subset of vector space V?

Yes, a singular matrix can be a proper subset of vector space V. A proper subset is a subset that does not contain all the elements of the original set. In this case, the singular matrix may not contain all the elements or fulfill all the requirements of vector space V, making it a proper subset.

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