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 ( ax^{4}+bx^{3}+cx^{2}+dx+e) with the usual operations;
(d) the set of polynomials of degree
with the highest coefficient
1 (
x^{4}+ax^{3}+bx^{2}+cx+d) and
the usual operations;
(e) the set of all pairs of real numbers (x,y) with the
operations
(f) the set of all pairs of positive real numbers (x,y) with the
operations
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
(ax^{2}+bx+c)
( a) 1, x^{2}, x^{2}-2 | ( c) x+2, x+1, x^{2}-1 | |
( b) 2, x^{2}, x, 2x+3 | ( d) x+2, x^{2}-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.