1. Determine which sets are vector spaces under the given operations. For those which are not, explain why.
(a) the set of triangular matrices with the usual addition and scalar multiplication;
(b) the set of matrices having the form with the usual addition and scalar multiplication;
(c) the set of polynomials of degree ( ax4+bx3+cx2+dx+e) with the usual operations;
(d) the set of polynomials of degree
with the highest coefficient
the usual operations;
(e) the set of all pairs of real numbers (x,y) with the
(f) the set of all pairs of positive real numbers (x,y) with the
2. Determine whether or not the following vectors span .
|( a) (1,2) and (-1,1)||( c) (-2,1), (1,3) and (2,4)|
|( b) (2,3) and (4,6)||( d) (-1,2), (1,-2) and (2,-4)|
3. Which of the following polynomials span the vector space of polynomials of degree (ax2+bx+c)
|( a) 1, x2, x2-2||( c) x+2, x+1, x2-1|
|( b) 2, x2, x, 2x+3||( d) x+2, x2-1|
4. Let and . Determine whether or not the following vector belongs to (if ``yes" represent as a linear combination of and )
|( a)||( b) .|
5. Let U and V be subspaces of a vector space W. Prove that is also a subspace of W.