Question

Let K be the vector space. Show that X is a subspace.

Answer 1

To show W is a subspace, you need to show three things.
1. \(0\in W\)
2. If \(B,C\in W\), then \(B+C\in W\).
3. If \(a\) is a scalar, and \(B\in W\), then \(cB\in W\).

Answer 2

The 10x10 part isn't really that important. This proposition is true for for any vector space.

Answer 3

I think you can show the first condition on your own. As for the next, let \(B,C\in W\). So \(AB=BA\) and \(AC=CA\). Then look at \(A(B+C)\). This is equal to \[AB+AC=BA+CA=(B+C)A.\]So \(A(B+C)=(B+C)A\), so \(B+C\in W\).

Answer 4

Correct. Although, if you've been told another way to check if something is a subspace, you might want to try it that way or your teacher may not like your solution.

Answer 5

okay thank you :)

Answer 6

You're welcome.

Answer 7

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