Linear Algebra

MATH 413-513:

TEST 2

MAKE-UP

SPRING 2014

Directions: The rules for the make-up are as follows: You may either complete the make-up and turn it in on time (it is due Monday, April 21 before class starts), or not. If you choose not to turn in the make-up, then your grade on Test 2 will stay the same. If you turn the make-up in, I will grade it and I will replace your grade on Test 2 by the average of your original score on Test 2 and your score on this make-up. Note: It is possible for your grade to go down.

1

2 SPRING 2014

(1) Let V be a finite dimensional vector space over F. Prove the following:

a) Let {v1, v2, · · · , vn} ⊂ V be a linearly independent collection of vectors in V . Show that the collection {w1, w2, · · · , wn}⊂ V with

w1 = v1, w2 = v1 + v2, and wi = v1 + v2 + · · · + vi for all 3 ≤ i ≤ n. is also a linearly independent collection of vectors in V .

b) Let {v1, v2, · · · , vn} ⊂ V be a linearly independent collection of vectors in V . Suppose that v ∈ V and v /∈ span(v1, v2, · · · , vn). Show that the collection {w1, w2, · · · , wn}⊂ V with

wi = vi + v for all 1 ≤ i ≤ n. is also a linearly independent collection of vectors in V .

MATH 413-513: TEST 2 MAKE-UP 3

(2) a) Let V , W , and U all be finite dimensional vector spaces over the same field F. Suppose T ∈ L(V, W ) is injective and S ∈ L(W, U) is surjective. If dim(V ) = m, dim(U) = n, and range(T ) = null(S), then find dim(W ). Show your work.

b) Let V and W be vectors spaces over the same field F. Let T ∈ L(V, W ). We say that T has a left inverse if there is a mapping L ∈L(W, V ) for which:

LT = IV

where IV ∈L(V ) is the identity mapping on V . We say that T has a right inverse if there is a mapping R ∈L(W, V ) for which:

TR = IW

where IW ∈L(W ) is the identity mapping on W . Prove the following:

i) If T has a right inverse, then T is surjective.

ii) If T has a left inverse, then T is injective.

4 SPRING 2014

(3) The goal of this exercise is to prove an analogue of Theorem 7.5.3 for lower triangular matrices. The proof will be similar to what we did in class (also it will be similar to what is in the book), but I want you to write it all out in your own words; if you skip parts, you will lose points.

a) Define lower triangular. (See Definition 7.5.1).

b) Prove the appropriate analogue of Proposition 7.5.2 for a basis in which M(T ) is lower triangular.

c) Prove the appropriate analogue of Theorem 7.5.3, i.e. there exists a basis in which M(T ) is lower triangular.

MATH 413-513: TEST 2 MAKE-UP 5

(4) Let T : C4 → C4 be given by T (x) = Ax for all x ∈ C4

where the matrix A is

A =

 

i 3 2 + i 11 0 1/2 3 − i 8 0 0 0 4 0 0 0 −2i

  .

a) Find the eigenvalues and eigenvectors of T . State any results you use in doing so.

b) Let B be the collection of eigenvectors found in part a). Show that B is a basis of C4 and find the matrix corresponding to T in this basis (as both domain and range basis). State any results you use in doing so.