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Matrices, Vector Spaces and Subspaces

Give a demonstration as to why or why not the given objects are vector subspaces of M22

a) all 2 X 2 matrices with integer entries

A vector space is a set that is closed under finite vector addition and scalar multiplication.

It is not a vector space, since V is NOT closed under finite scalar multiplication. For instance, take a 2 by 2 matrix with integer entries

Then is NOT in V since not every entry is an integer.

b) all matrices a b

c d such that: a + b + c + d = 0

It is a vector space, since V is closed under finite vector addition and scalar multiplication.

(1) V is closed under finite vector addition

For instance, take any two 2 by 2 matrices in V

and
where and . Obviously, we have
=
as

And
(2) V is closed under finite scalar multiplication.

as

c) all 2 X 2 matrices A such that det(A) = 0

It is NOT a vector space, since V is NOT closed under finite vector addition. For instance, take TWO 2 by 2 matrices with det(A)=det(B)=0

and
BUT
with det(A+B)=2*2-1*1=3
So,

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