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Math 010 Exam #1 Chapters 0 – 1.5 Name:__________________________________

 

Do all your work in the blue book. No notes or outside help. Approved calculators are allowed(but are

entirely unnecessary.) Proofs are marked with a *

 

1. Justify every step by naming the appropriate field axiom.

A)* (6 pts) Prove the uniqueness of additive inverses: If x, w are elements of R, then

x + w = 0 if and only if w = -x. Be sure to show both directions, even if one direction is trivial.

B) (2 pts) State carefully, in words, using “additive inverse,” “multiplicative identity,” etc., the meaning

of (-1)*x = -x.

C)* (6 pts) Use A) and the field axioms to prove that (-1)*x = -x, for all x in R. You may also use the zero

factors (annihilator) theorem.

D) (Bonus – 3 pts) Demonstrate the result in part “C” using Z/7Z and x = 2.

 

2. A) (2 pts) Define the dot product, �⃗� ∘ 𝑣 , of two vectors �⃗� and 𝑣 in Rn.

B)* (6 pts) Prove, using the field axioms for R and your definition in A), that for all �⃗� , 𝑣 in Rn and for all s

in R, s(�⃗� ∘ 𝑣 ) = (s�⃗� ) ∘ 𝑣 .

3. (14 pts total)

A) State the Cauchy-Schwarz inequality for vectors in Rn.

B) Define the cosine of the angle between any two vectors in Rn.

C) Explain why the Cauchy-Schwarz inequality guarantees that the definition in B) makes sense.

D)* Use Cauchy-Schwarz to prove the Triangle Inequality.

 

4. (12 pts total) Consider the system:

 

2x + y – 3z = 5

5x + 2y – 8z = 13

2x + 2y – 2z = 4

 

A) Form the augmented matrix for the system.

B) Find the RREF of the augmented matrix. Show each elementary row operation you used.

C) Interpret your result in terms of the original system. If there are an infinite number of solutions,

express them in terms of the parameter, t.

D) Interpret your result geometrically in R3

5. (10 pts total)

A) Find an equation for the plane through the origin spanned by �⃗� =<-2, 1, 1> and 𝑣 =<3, 0, -2>.

B) Find a nonzero vector �⃗⃗� that is orthogonal to �⃗� , such that Span{�⃗� , �⃗⃗� } = Span{�⃗� , 𝑣 }

 

6. (12 pts total) A) Define what it means for two lines to be parallel in R3.

B) Show that L1: <x, y, z> = <3, 0, 2> + t <-6, 2, -4> and L2: <x, y, z> = <-2, 1, 5> + s <3. -1. 2> are parallel.

C) Find an equation for the plane in R3 containing L1 and L2.

 

 

 

 

7. (4 pts total) If S is a set of two or more vectors from Rn, define

A) A linear combination of vectors from S.

B) Span (S)

 

8. Label each statement as either “true” or “false.” Explain your choice, or give an example (10 pts total)

A) An underdetermined linear system has an infinite number of solutions.

B) A homogeneous system of 3 equations and 4 unknowns has non-trivial solutions.

C) A set of 8 non-zero vectors from R5 is always linearly dependent.

(16 pts total) Below are three sets of vectors from R4 and three vectors from R4. In each case:

A) Form the augmented matrix, A, you would use to determine if �⃗� is in Span(S)

B) The RREF of the correct augmented matrix from A) is given. Use it to determine whether:

i) �⃗� is not in Span(S)

ii) �⃗� has a unique representation in vectors from Span(S)

iii) �⃗� has an infinite number of representations in vectors from Span(S).

 

In Case i) Explain how you know.

In Case ii) Find x1, x2 and x3 such that �⃗� = 𝑥1𝑣1⃗⃗⃗⃗ + 𝑥2𝑣2⃗⃗⃗⃗ + 𝑥3𝑣3⃗⃗⃗⃗ , and show that it works.

In Case iii) a) Identify the free variable. b) Find the simplest solution to the equation (using the fewest

vectors) and show that it works for the original vector set. c) If we add the condition that x1 = x3, find the

unique solution and show that it works for the original set.

 

9. S = { <2, -1, 3, 1>, <3, 7, 3, -1>, <-2, 4, -2, 1>} , �⃗� = <1, 14, 2, 2>

 

RREF(A) = (

1 0 0 1 0 1 0 1 0 0 1 2 0 0 0 0

)

 

10. S = {<2, -1, 3, 1>, <3, 7, 3, -1>, <1, -9, 3, 3>} , �⃗� = <3, 1, 2, 5>

 

RREF(A) = (

1 0 2 0 0 1 −1 0 0 0 0 1 0 0 0 0

)

 

11. S = {<2, -1, 3, 1>, <3, 7, 3, -1>, <1, -9, 3, 3>} , �⃗� = <0, -17, 3, 5>

 

RREF(A) = (

1 0 2 3 0 1 −1 −2 0 0 0 0 0 0 0 0

)

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