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I am brand new to learning about subspaces in my linear Algebra class. Ive tried to follow khan academy but to no avail. I encountered these practice problems in the textbook. However my textbook inconveniently provides answers for odd numbered problems only.

I want to further my understanding. Could someone please explain how to go about these problems?

Given vector space V = the set of all 2 × 2 matrices.....

Does the set of all 2 × 2 nonsingular matrices form a subspace of
V? Explain.

Does the set of all 2 × 2 singular matrices form a subspace of V?
Explain.

Thank you :)

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  • $\begingroup$ Always assume it is not a subspace first. Can you think of properties which might not hold for particular elements in these sets? $\endgroup$ – David Peterson Oct 13 '16 at 4:38
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To get you started... Let $\operatorname{GL}_2$ denote the set of $2 \times 2$ nonsingular matrices. The identity $I$ is in $\operatorname{GL}_2$... but what about $I - I = 0$?

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Hint: (that is surprisingly familiar to you)

  • Can you find two nonsingular matrices that sum to a singular matrix?
  • Can you find two singular matrices that sum to a nonsingular matrix?

What do these imply?

Try to write down what are the conditions that you have to verify to show that something is a subspace.

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