Determining Subspace of Singular 2x2 Matrices in R^2

[3.141592654]

Homework Statement

Determine if the set of all singular 2 x 2 matrices are a subspace of $R^{2}$

Homework Equations

If a, b, c, and d are the entries of a 2 x 2 matrix, then their determinant, $ad - bc = 0$ if the matrix is singular.

The Attempt at a Solution

I have been doing other problems like this without much trouble by taking two general form objects in the set and checking the closure relations for addition and scalar multiplication. However, I'm not sure how I can represent the general form of a singular 2 x 2 matrix so that I can perform addition and scalar multiplication on it.

For example, the matrix $X$ with $x_{11}=a, x_{22}=b, x_{21}=a,$ and $x_{22}=b$ is an example of a singular 2x2 matrix but it isn't a general form, which I'd need to determine if the set of singular 2x2 matrices are a subspace.

If I took two matrices like x and checked the closure relations I believe I'd find they are satisfied, which my books tells me is the wrong answer.

Thanks.

 

[3.141592654]

I have another similar question and I'm hoping I can't get help with both of them. The 2nd question is:

Homework Statement

Determine whether the set of all polynomials in $P_{4}$ (where $P_{4}$ is the set of polynomials with degree less than 4) having at least one real root is a subspace of $P_{4}$.

The Attempt at a Solution

Like the problem above, I'm just trying to figure out how to represent the set of all polynomials in $P_{4}$ having at least one real root. I'm not sure where to begin and am looking for help with the start. Thank you.

 

[Dick]

Homework Statement

Determine if the set of all singular 2 x 2 matrices are a subspace of $R^{2}$

Homework Equations

If a, b, c, and d are the entries of a 2 x 2 matrix, then their determinant, $ad - bc = 0$ if the matrix is singular.

The Attempt at a Solution

I have been doing other problems like this without much trouble by taking two general form objects in the set and checking the closure relations for addition and scalar multiplication. However, I'm not sure how I can represent the general form of a singular 2 x 2 matrix so that I can perform addition and scalar multiplication on it.

For example, the matrix $X$ with $x_{11}=a, x_{22}=b, x_{21}=a,$ and $x_{22}=b$ is an example of a singular 2x2 matrix but it isn't a general form, which I'd need to determine if the set of singular 2x2 matrices are a subspace.

If I took two matrices like x and checked the closure relations I believe I'd find they are satisfied, which my books tells me is the wrong answer.

Thanks.

I don't think the sum of two singular matrices is necessarily singular. Can you find an example?

 

[Dick]

I have another similar question and I'm hoping I can't get help with both of them. The 2nd question is:

Homework Statement

Determine whether the set of all polynomials in $P_{4}$ (where $P_{4}$ is the set of polynomials with degree less than 4) having at least one real root is a subspace of $P_{4}$.

The Attempt at a Solution

Like the problem above, I'm just trying to figure out how to represent the set of all polynomials in $P_{4}$ having at least one real root. I'm not sure where to begin and am looking for help with the start. Thank you.

Same strategy on this one. Find two polynomials with at least one root whose sum has no roots. Just flail around for a bit, you'll probably find two now that you know you can.