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#1. Let ? be the set of all diagonal matrices in ?22. Is ? a subspace...

#1. Let ? be the set of all diagonal matrices in ?22. Is ? a subspace of ?22? Justify your conclusion (write a proof).

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Answer #1

1. Let A =

a

0

0

b

and B =

c

0

0

d

where a,b,c,d are arbitrary real numbers ,be 2 arbitrary diagonal matrices inW, and let k be an an arbitrary real scalar.

Then A+B =

a+c

0

0

b+d

Now, A+B being a 2x2 diagonal matrix, belongs to W . Hence, W is closed under vector addition.

Also, kA =

ka

0

0

kb

Now, kA being a 2x2 diagonal matrix, belongs to W . Hence, W is closed under scalar multiplication.

Further, the 2x2 zero matrix, being a diagonal matrix, belongs to W .

Hence W is a vector space, and, therefore, a subspace of M22.

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