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Which of the following subsets are subspaces of the space of 2 × 2 matrices under the usual matrix operations.

  1. The set of matrices of the form $$ \begin{bmatrix} a & -a \\ b & 0 \end{bmatrix} $$ where a and b are arbitrary real numbers.
  2. The set of symmetric 2 × 2 matrices.
  3. The set of singular 2 × 2 matrices.

and also

Which of the following subsets are subspaces of the space of polynomilas of degree at most 5.

  1. The set of polynomials for which the sum of all coefficients is equal to zero.
  2. The set of polynomials f(x) such that f(2) = f(3).
  3. The set of polynomials f(x) such that f'(2) = 1.

I know barely anything about subspaces so i'm not sure how to answer these (even though i'm sure its fairly easy considering its our first homework over this)

Here's what I think:

First question: 1 and 2 are subspaces, but 3 isn't because its not closed under addition.

Second question: 1 and 2 are also subspaces but 3 isn't because its not closed under addition or scalar multiplication.

Am I right?

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closed as off-topic by uniquesolution, C. Falcon, Juniven, Namaste, Daniel W. Farlow Mar 6 '17 at 0:42

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  • $\begingroup$ Hint : Check, whether every product of matrices with a given property again has the given property. $\endgroup$ – Peter Mar 5 '17 at 21:03
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    $\begingroup$ @Peter I think I think you mean the sum of matrices. $\endgroup$ – pjs36 Mar 5 '17 at 21:08
  • $\begingroup$ @Peter can you explain what you mean in that hint? $\endgroup$ – Jeg Mar 5 '17 at 21:08
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    $\begingroup$ I'm voting to close this question as off-topic because it is another do-my-homework post. $\endgroup$ – uniquesolution Mar 5 '17 at 21:12
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    $\begingroup$ @Jeg I've edited your comment into the question. Generally people on the site like to know the person has attempted a problem, or spent time trying to articulate what is causing them trouble. $\endgroup$ – pjs36 Mar 5 '17 at 22:41

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