Homework 2 Algebra Mathematics Assignment Help

Homework 2 Algebra Mathematics Assignment Help

HOMEWORK 2
DUE MONDAY APRIL 16TH AT 2PM
Your homework should be written on standard-sized paper, and loose sheets
must be stapled together.
If no operations are specified for a vector space, you may assume that the addition and
scalar multiplication are the standard ones given in class.
(1) Let V = R, with a new addition
and scalar multiplication · defined by
a
b := a + b − 2;
λ · a := λa − 2λ + 2.;
for all a, b ∈ V and all λ ∈ R. (The addition and multiplication on the right
hand side of each equality is the usual addition and multiplication in R). You
may assume that
and · are commutative and associative.
Show that V is a vector space over R with respect to
and ·.
(2) For each of the following, decide whether it is a subspace of R
R or not. Justify
your answer in each case (either prove it is a subspace, or give an example to
show how it fails).
(a) {f ∈ R
R | f(0) = 0}.
(b) {f ∈ R
R | f(0) ∈ Z}.
(c) {f ∈ R
R | f is a polynomial of degree 2}
(d) {f ∈ R
R | f is differentiable}
(e) {f ∈ R
R | f is even, i.e. f(−x) = f(x) for all x ∈ R}
(3) Let m, c ∈ R. Show that the set of points (x, y) that lie on the straight line
y = mx + c form a subspace of R
2
if and only if c = 0. (Hint: draw it!)
(4) Prove that if W1 and W2 are both subspaces of a vector space V over a field F,
then W1 ∩ W2 is also a vector space over F.
(5) Let W1 = {(a, b, a) | a, b ∈ R} and W2 = {(0, 0, c) | c ∈ R} be subspaces of
R
3
. What is W1 + W2? Is the sum direct? Give a justification for each of your
answers.
1