Determine whether the set, together with the indicated operations, is a vector s

Question

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 3 x 3 matrices of the form b co with the standard operations The set is a vector space. O The set is not a vector space because it is not closed under addition O The set is not a vector space because an additive identity does not exist. O The set is not a vector space because an additive inverse does not exist O The set is not a vector space because it is not closed under scalar multiplication.

Explanation / Answer

1.      The set is a vector space as it satisfies all the conditions for a vector space. On addition of any 2 matrices of the given form, the sum will be another matrix of the same form. Also, a,b,c,d,e,f are arbitrary real numbers. When these are all 0, the result is a zero matrix which is the additive identity in the given set. When e change the sign of each of a,b,c,d,e,f, the result is the additive inverse of the original matrix. On scalar multiplication, the result is a matrix of the same form as all the entries get multiplied by this scalar. Thus the set is closed under multiplication.

2.      The equations of 2 arbitrary quadratic functions whose graphs pass through (0,3) are f(x) = ax2 +3 and g(x) = bx2 +3. Then (f+g)(x)=f(x) +g(x) = (a+b)x2 +6. The graph of f(x) +g(x) does not pass through (0,3) ( It passes through( 0,6). Hence the set is not closed under vector addition so that it is not a vector space.

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