Please help me with the following Linear Algabra question. Please show work. Det
ID: 3034897 • Letter: P
Question
Please help me with the following Linear Algabra question. Please show work.
Determine whether the set, together with the indicated operations, is a vector space. If it is not, then identify one of the vector space axioms that fails. The set of all 2 times 2 matrices of the form [a c b 0] with the standard operations The set is a vector space. The set is not a vector space because it is not closed under addition. The set is not a vector space because an additive inverse does not exist. The set is not a vector space because it is not closed under scalar multiplication. The set is not a vector space because a scalar identity does not exist.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