Determine whether the following set, together with the indicated operations, is

Question

Determine whether the following set, together with the indicated operations, is a vector space. The set of all 2 times 2 matrices of the form [a c b 0] with the standard operations, where a, b and c are real numbers. The set is a vector space. The set is not a vector space. (b) Determine whether the following set, together with the indicated operations, is a vector space. The set of all 8 times 8 nonsingular matrices with the standard operations The set is a vector space. The set is not a vector space.

Explanation / Answer

[ a b , c 0 ]

It follows closure under addition : [ a1 b1 , c1 0 ] + [ a2 b2 , c2 0 ] = [ a1+a2 b1+ b2 , c1+ c2 0 ]

Additive inverse exists: [ a b , c 0 ] + [- a -b , -c 0 ] = [ 0 0 , 0 0 ]

closed under scalar multiplication : x[ a b , c 0 ] = [ xa , xb , xc , 0 ]

Set is a vector space

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