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LAG ProblemSheet3

1) Students are encouraged to work with others on problem sets and ask tutors for help if needed. 2) The problem set contains 7 multi-part problems involving linear algebra concepts like vectors, linear combinations, spans, angles between vectors, parametric equations of lines, orthogonal vectors, and the triangle inequality. 3) Students are asked to perform calculations, determine if expressions are defined, express vectors as linear combinations, find angles between vectors, find parametric equations of lines, determine if vectors are in a span, prove properties like the triangle inequality, and more.

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Dominic Wynes-Devlin
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65 views2 pages

LAG ProblemSheet3

1) Students are encouraged to work with others on problem sets and ask tutors for help if needed. 2) The problem set contains 7 multi-part problems involving linear algebra concepts like vectors, linear combinations, spans, angles between vectors, parametric equations of lines, orthogonal vectors, and the triangle inequality. 3) Students are asked to perform calculations, determine if expressions are defined, express vectors as linear combinations, find angles between vectors, find parametric equations of lines, determine if vectors are in a span, prove properties like the triangle inequality, and more.

Uploaded by

Dominic Wynes-Devlin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Linear Algebra and Geometry I 2021 Problem Sheet 3

You are encouraged to work with other students on the module (especially those in your
study group). If you are having difficulty with any of the questions or want feedback on
specific answers you should ask your tutor in your next tutorial or attend the lecturer’s
office hours.

1. Let u, v, w be vectors in Rn and let α be a scalar. State whether each of the following
operations are defined. If you think it is not defined, explain why.

(a) 3α + v (c) u/v (e) kukv − αu


(b) w + α2 v (d) kvk − αu (f) (u · v) · w


      
2 −3 0 4
2. Let v1 = −1 , v2 =
   5 , v3 = 0 , v4 = −3 .
   
1 0 6 −1

(a) Compute the following linear combinations:


1 1
(i) 3
(3v1 − 4v2 + 2v3 ) (ii) 2
(v2 − v1 ) + 4(v4 − v1 )

(b) Find x ∈ R3 such that 2v1 − v2 − x = 3(x + 2v4 ).

3. Let      
−1 + 2i 0 −i
u =  2i  , v =  i , w = 3 .

1 1−i 4

(a) Evaluate the following expressions.


u+v √
(i) 3u − iv + 2w (ii) (iii) ( 2eiπ/4 u + v) · w
ku + vk

(b) Determine whether each of the following vectors is contained in span{v, w}. If
it is, express it as a linear combination of v and w. (Recall that span{v, w}
denotes the linear span of v and w.)
     
2 − 3i 1 + 7i −3i
(i)  9 + 6i  (ii) 2 − 3i (iii)  12 + i 
12 + 8i 2 + 3i 10 − 4i

1
4. Let u = (1, −2, 3), v = (3, 0, 1) and w = (−2, 2, 1).

(a) Determine the cosine of the angle between

(i) u and v (ii) v and w

(b) Find α ∈ R such that u + αv is perpendicular to w.

5. (a) Find the parametric equation of the line in R3 passing through the point (4, 1, 5)
and parallel to the vector (1, 0, 1).
(b) Find the parametric equation of the line in R3 passing through the points
(2, −7, 12) and (2, 9, −6).
(c) Show that the lines in Parts (a) and (b) intersect, and find their point of inter-
section.
(d) Let L1 and L2 be the lines in R3 with parametric equations
       
0 2 1 1
L1 : v = 1 + t 2
    and L2 : v = 2 + t 1 .
  
1 2 2 1

Show that both lines pass through the points (0, 1, 1) and (1, 2, 2). Does that
mean L1 and L2 are equal? Explain your answer.

6. Let p = (−2, 1, −3) and q = (0, 4, 1).

(a) Show that the vector (13, 2, −8) is orthogonal (i.e perpendicular) to both p and
q.
(b) Hence determine the Cartesian equation of the plane with parametric equation
 
1
v=  5  + tp + sq.
−2

7. Prove the triangle inequality in Rn , i.e. prove that for every u, v ∈ Rn we have

ku + vk ≤ kuk + kvk.

You may use the fact that |u · v| ≤ kukkvk.


(Hint: Express the length of u + v using the dot product.)
Make sure you write in sentences, with punctuation, and you explain what you are
doing. Look at the other proofs in the notes for guidance on how to write a proof.

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