Linear Algebra 2 — Homework 1
Due at the beginning of class on Friday, March 12
Instructions: Do your work on scratch paper and transfer final solutions to another paper
which will be handed in. Make sure that your solutions are in the correct order and very,
very neat. Proofs must be written correctly.
1) Let x + 2y + 3z = 0 be the plane P in R3 .
a) Quickly find two vectors u and v so P = {αu + βv : α, β ∈ R} (we say that P is generated
by u and v or that P is the linear span of u and v). I do not want these two vectors to be
perpendicular.
b) Let A be the 3 × 2 matrix whose columns are the two vectors u and v you found in (a).
Form the matrix A(A> A)−1 A> .
c) Now take a little more time and find u and v so that u⊥v (but don’t use unit vectors)
and P = {αu + βv : α, β ∈ R} .
d) Let A be the 3 × 2 matrix whose columns are the two vectors u and v you found in (c).
Form the matrix A(A> A)−1 A> .
e) Now modify (c) so that u and v are orthonormal (this is, in addition to being orthogonal,
u and v are also unit vectors).
f) Let A be the 3 × 2 matrix whose columns are the two vectors u and v you found in (e).
Form the matrix A(A> A)−1 A> .
2) Let A ∈ Mm,n (R). Prove that if x ∈ Rn and y ∈ Rm , then Ax · y = x · A> y. (Hint: you
might want to observe that if u and v are vectors, then u · v = v> u. You might also want to
remember that a matrix (or vector) transposed twice will yield the same matrix. You might
also want to remember how to take the transpose of a product).
3) Let A ∈ Mm,n (R). Prove
a) N (A) ⊂ N (BA) for any k × m matrix B
b) Give an example to show that N (BA) ⊂ N (A) is not always true.
c) Prove that N (A> A) ⊂ N (A).
d) Put (a) and (c) together to conclude that N (A> A) = N (A).
4) Let {v1 , v2 , ..., vk } be the columns of a n × k matrix A (thus all the vectors vi are in Rn ).
Then prove the following statements are equivalent.
a) N (A) = {0}
b) {v1 , v2 , ..., vk } is a linearly independent set of vectors
c) A> A is invertible
5) Number 5 on page 37.
6) Number 2 on page 42. Also describe the trace of A> A.
7) Number 1 on page 61.
6) Number 4 on page 61