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Tutorial 5

This document is a tutorial for the B.Tech. II Semester Engineering Mathematics II course at Amrita School of Engineering, focusing on inner product spaces and orthogonality. It includes problems related to inner products, vector evaluations, unit circle sketches, cosine of angles, orthogonal vectors, and verification of the Cauchy-Schwarz inequality. Additionally, it poses true/false questions regarding orthonormal systems and linear independence.

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kenneth kilian
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13 views1 page

Tutorial 5

This document is a tutorial for the B.Tech. II Semester Engineering Mathematics II course at Amrita School of Engineering, focusing on inner product spaces and orthogonality. It includes problems related to inner products, vector evaluations, unit circle sketches, cosine of angles, orthogonal vectors, and verification of the Cauchy-Schwarz inequality. Additionally, it poses true/false questions regarding orthonormal systems and linear independence.

Uploaded by

kenneth kilian
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Amrita School of Engineering

B.Tech. II Semester
23MAT130 - Engineering Mathematics II
Tutorial 5
(Inner product space, orthogonality)

1. Let 𝑢 = (𝑢1 , 𝑢2 , 𝑢3 ), 𝑣 = (𝑣1 , 𝑣2 , 𝑣3 ). Determine whether the given function 〈𝑢, 𝑣〉 = 𝑢1 𝑣1 + 𝑢3 𝑣3


is an inner product on 𝑅 3 . If not, list the axioms that do not hold.
2. Suppose that 𝑢, 𝑣 and 𝑤 are vectors such that
〈𝑢, 𝑣〉 = 2, 〈𝑣, 𝑤〉 = −3 , 〈𝑢, 𝑣〉 = −3, 〈𝑢, 𝑤〉 = 5, ‖𝑢‖ = 1, ‖𝑣‖ = 2, ‖𝑤‖ = 7 .
Evaluate the given expression
a) 〈𝑢 − 𝑣 − 2𝑤, 4𝑢 + 𝑣〉
b) ‖𝑢 − 2𝑣 + 4𝑤‖
3. Sketch the unit circle in 𝑅 2 using the given inner product
1 1
a) 〈𝑢, 𝑣〉 = 9 𝑢1 𝑣1 + 25 𝑢2 𝑣2
1 1
b) 〈𝑢, 𝑣〉 = 25 𝑢1 𝑣1 + 16 𝑢2 𝑣2
𝑐) 〈𝑢, 𝑣〉 = 2𝑢1 𝑣1 + 𝑢2 𝑣2
4. Let 𝑅 3 have the Euclidean inner product. Let 𝑢 = (1, 1, −1), 𝑣 = (6,7, −15).
a) If ‖𝑘𝑢 + 𝑣‖ = 13, what is 𝑘?
b) Find 𝑑(𝑢, 𝑣)
5. Find the cosine of the angle between 𝑢 𝑎𝑛𝑑 𝑣 with respect to appropriate inner product spaces.
a) 𝑢 = (1, 0, 1, 0), 𝑣 = (−3, −3, −3, −3) in 𝑅 4
b) 𝑝 = 1 − 𝑥 + 2𝑥 2 , 𝑞 = 2𝑥 + 𝑥 2 in 𝑃2
9 8 7 1 2 3
c) 𝐴 = [ ] and 𝐵 = [ ] in 𝑀2𝑥2 where 〈𝐴, 𝐵〉 = 𝑡𝑟(𝐵 𝑇 𝐴)
6 5 4 4 5 6
6. Find a non-zero vector 𝑤 that is orthogonal to 𝑢1 = (1, 2, 1) and 𝑢2 = (2, 5, 4) in 𝑅 3 .
7. For what values of 𝑘 the vectors are orthogonal?
a) 𝑢 = (𝑘, 𝑘, 1), 𝑣 = (𝑘, 5, 6) in 𝑅 3
2 1 1 1
b) 𝐴 = [ ], 𝐵 = [ ] in 𝑀2𝑥2 where 〈𝐴, 𝐵〉 = 𝑡𝑟(𝐵 𝑇 𝐴)
−1 3 𝑘 −1
8. Verify the Cauchy-Schwarz inequality using the Euclidean inner product for the vectors 𝑢 = (3, 2),
𝑣 = (4, −1) where Cauchy-Schwarz inequality is |〈𝑢, 𝑣〉| ≤ ‖𝑢‖ ‖𝑣‖.
0 4/5 −3/5
9. Check whether S={(0) , (3/5) , ( 4/5 )} forms an orthonormal basis for R3. If yes, express (1, -1, 2),
1 0 0
(3, 5, 7) and (2, 0, 8) as linear combinations of the elements in S.
10. State True or False:
a. If all row vectors of a matrix form an orthonormal system, then the column vectors of the matrix also
will form an orthonormal system.
b. If S is a set of orthonormal vectors, then S will also be a linearly independent set.

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