linear algebra questions
Math 2568, Sec 001 Spring 2020
Homework 4 Due Thursday, March 19, 2020
Instructions:
• Complete each of the problems to the best of your ability. Show all work that leads to your final answer. Any explanations or justifications should be written out in full sentences and (reasonably) correct grammar.
• Only submit your final product. Scratch work should be worked separately and then recopied neatly onto standard letter-sized paper before submission. Your assignment should be stapled with your name clearly labelled on each page. Your work should be legible and the problems should be in the correct order. Do not make me hunt for problems or their supporting work!
Calculator Use:
You may use the ref and rref functions on your calculator to perform any row reduction. You must write the original matrix and the resulting echelon form as part of your work.
Problems:
The following problems are related to the material in Sections Two.II.1 and Two.III.1-2 .
1. Determine whether the following sets are linearly independent. Show your work.
(a) S = {( 2
1
) ,
( 4
0
) ,
( 9
8
) ,
( −1 5
)} (b) S =
{( 1 0
1 2
) ,
( 1 2
2 1
) ,
( 0 1
2 2
)}
2. Consider the following two bases on R2, based on of the allowable moves of the bishop (B) and the knight (K) on a chessboard:
B =
⟨( 1
1
) ,
( −1 1
)⟩ Bishop
K =
⟨( 2
1
) ,
( −1 2
)⟩ Knight
Math 2568, Sec 001 Spring 2020
(a) Identify the vector ~u ∈R2 whose representation with respect to B is RepB (~u)= ( 5
−2
) .
(b) Identify the vector ~v ∈R2 whose representation with respect to K is RepK (~v)= ( −3 8
) .
(c) Let ~w = ( 7
1
) . Find RepB (~w) and RepK (~w).
3. The first four Laguerre polynomials are:
1, 1− t, 2−4t+ t2, 6−18t+9t2 − t3
(a) Show that the set of the first four Laguerre polynomials is linearly independent.
(b) Explain why the first four Laguerre polynomials form a basis B of P3.
(c) Using the basis from (b), find coordinates of p(t)=−10t+9t2 − t3 relative to B.
(d) Using the basis from (b), identify the polynomial q(t)∈P3 whose representation with respect to
B is RepB (q)=
1
0
−5 2
.
4. Let U be the subspace of M2×2 consisting of the following matrices:
U =
{( a b
c d
)∣∣∣∣ a+d =0 }
Find a basis for U and determine the dimension of U.
5. Let W be the subspace of P3 consisting all odd third degree polynomials. Find a basis for W and determine the dimension of W. Recall that a function f is odd if f (−x)=−f (x).
6. Give an example of the following or explain why no such example exists:
(a) a basis for M2×2 that contains ( 0 0
0 0
) (b) a basis of P3 that contains exactly 3 polynomials
(c) two nonzero vectors in R3 that are linearly dependent