Give me all steps and anwser. 

Math 2568, Sec 001 Spring 202

0

Homework 3 Due Tuesday, March 3, 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 Section Two.I.2.

1. Let W be the subspace of P3 consisting of all third degree polynomials p(x) such that p(−1)= p(1)=0.

(a) Determine whether f (x)=4+x −4×2 −x3 is in W.

(b) Verify that W contains the the ‘zero vector’ of P3.

(c) Show W is closed under addition and scalar multiplication.

(d) Find a set of third degree polynomials that spans W. That is, parametrize W.

2. Let V be the subset of R2 given by

V =

{(
x

0

) ∣∣∣∣ x is an even integer
}

Using the standard notion of addition and scalar multiplication of 2-dimensional vectors:

(a) Show that V is closed under addition.

(b) Show that V is not subspace of R2.

3. (a) Consider the homogeneous linear system represented by the following augmented matrix:(
1 −3 2 0
0 0 3 0

)

Show that the solution set SH to this system is a subspace of R3.

Math 2568, Sec 001 Spring 2020

(b) Consider the related nonhomogeneous system:(
1 −3 2 8
0 0 3 1

)

Determine whether the solution set SN of the nonhomogeneous is a subspace of R3. Explain.

In part (a) you showed that the solution set to the given homogeneous linear system is a subspace of
R3. In fact, the solution set to any homogeneous system is always a subspace:

Proposition 1: The solution set of any homogeneous linear system is a subspace of Rn, where n is the
number of variables in the system.

4. Give an example of each or explain why it it is impossible:

(a) A subspace of M2×3 that contains exactly one vector.

(b) A subspace of R2 that contains exactly two vectors.

(c) A nonempty subset of P1 that is not a subspace of P1.

5. Let S = {A,B,C} be a subset of M2×2 where:

A=

(
4 0

−2 −2

)
B =

(
1 −1
2 3

)
C =

(
0 2

1 4

)
(a) Determine whether each of the following matrices in the span of S.

i.
(
6 −8
−1 −8

)
ii.

(

0 0

0 0

)
iii.

(
−1 5
7 1

)

(b) Extra Credit: For each matrix in part (a) that is in the span of S, express the matrix as a linear
combination of A,B, and C.