HW for Linear Algebra (Need it in 6 hrs)

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 RR 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 ∈ RR | f(0) = 0}.

(b) {f ∈ RR | f(0) ∈ Z}.

(c) {f ∈ RR | f is a polynomial of degree 2}

(d) {f ∈ RR | f is differentiable}

(e) {f ∈ RR | 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 R2 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 R3. What is W1 + W2? Is the sum direct? Give a justification for each of your answers.

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