Problem 1. For the following, Evaluate whether the given sets of vectors form a basis or not. If you think it is a basis, then prove it, if not then show why not. Explain what you do.

(1) {(1, 3), (-2, 1)} for R²
(2) {(1, 2, 1), (2, 0,-1), (4, 4, 1)} for R³
(3) {(1, 2, 1), (2, 1,-1), (4, 4, 1)} for R³
(4) {(1, 0, 4, 0, 6, 7), (0, 2, 0, 0, 1,-3), (-2, 0, 0, 0, 3,-4), (0, 2, 3, 0, 4, 5), (-2, 7,-5, 0,-3, 9), (-1, 0, 3, 0, 4,-5)} for R⁶

Problem 2. Recall that the dot product in Rn is given by ?v,w? = v · w = v₁w₁ + v₂w₂ + . . . + v_nw_n, where v = (v₁, . . . ,v_n) and w = (w₁, . . . ,w_n). Two vectors v and w are perpendicular exactly when ?v,w? = 0. The length of a vector v is de?ned to be

An n × n matrix Q is orthogonal if Q^TQ = I, that is, Q^T = Q-¹, where superscript T denotes transpose.

(1) Show that Q is orthogonal if and only if the columns of Q form an orthonormal basis {Q , . . . ,Q }, that is,

(2) Show that a 2×2 orthogonal matrix Q has one of the following two forms.

(3) What do these two matrices do to vectors when you apply the matrix to the vector Qv? Choose several particular  and apply the matrices to vectors (using Maple if you wish) to see what effects these matrices have. Explain what you see in your examples. Finally, prove what you believe by doing the linear transformations to the standard basis vectors e1 = (1, 0) and e2 = (0, 1) and interpreting the output geometrically.

Problem 3. Prove the subsequent statements.

a.) A linear transformation T . Rm → Rm is invertible (i.e. T 1 exists) if and only if, for any basis {v1, . . . , vm} of Rm, {T(v1), . . . , T(vm)}
is also a basis of Rm.

b.) A linear map T . Rn → Rm is not one-to-one if m < n.

c.) A linear map T . Rn → Rm is not onto if m > n.

Problem 4. Let T . V → V be a linear operator on an n-dimensional vector space V . Show that the following statements are equivalent.

(1) T^-1 exists.

(2) T is a one-to-one mapping (i.e., T(x) = T(y) implies x = y). This is also called an injective mapping.

(3) N(T) = 0 (This is Meyer's notation for the kernel of T).

(4) T is an onto mapping (i.e., for each v ∈ V , there is an x ∈ V such that T(x) = v). This is also called a surjective mapping.

Problem 5. Using the torus example from class, compute the homology groups of the projective plane RP² which can be thought of as a planar disk of radius 1 whose antipodal points have been identi?ed. This space cannot exist in 3-space R³ without having self-intersections. It shows an immersion of RP² in R³ called Boy's surface in honor of its creator, Werner Boy. It shows a way to cut up RP² similar to what you did with the torus in class. Again, v₁ is identi?ed with v₁₁ etc and this holds for all edges between identi?ed vertices as well with the orientation eij goes from v_i to v_j. Find H₀(RP²), H₁(RP²), and H₂(RP²).