# MATH 33AH Lecture Notes - Lecture 13: Orthonormality, Qr Decomposition, Glossary Of Video Game TermsExam

by OC2716258

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MathematicsCourse Code

MATH 33AHProfessor

AllStudy Guide

FinalThis

**preview**shows half of the first page. to view the full**2 pages of the document.**MATH 33A/2 PRACTICE MIDTERM 2

Problem 1. (True/False) ...

Problem 2. Let v1=ï£®

ï£°

1

1

0

ï£¹

ï£», v2=ï£®

ï£°

1

0

1

ï£¹

ï£», v3=ï£®

ï£°

1

1

1

ï£¹

ï£».

(a) Show that B= (v1, v2, v3)is a basis for R3.

(b) Let u=ï£®

ï£°

1

âˆ’1

2

ï£¹

ï£». Find the coordinates of uwith respect to the basis B.

(c) Find the change of basis matrix from the standard basis to B.

(d) Let T:R3â†’R3be the reï¬‚ection about the plane x+y+z= 0. Find the matrix of Twith

respec to B.

(e) Let T:R3â†’R3be a linear transformation. Show that Tis invertible if and only if Bâ€²=

(u1, u2, u3), where u1=T v1,u2=T v2and u3=T v3, is a basis for R3.

(f) With the same notation as in (e), assume that Tis invertible. Express the change of basis matrix

from the standard basis to the basis Bâ€².

Problem 3. (a) Prove that if V, W are subspaces of Rnand Vis a subspace of W, then WâŠ¥is a

subspace of VâŠ¥. Prove that Rn=VâŠ¥+Vand that Vâˆ©VâŠ¥= 0. (b) Let VâŠ‚R5be a subspace,

and let Wconsist of the zero vector and also all vectors win R5that make the angle 30 degrees

with every vector in V. Is Wa subspace? Prove your answer.

Problem 4. (a) Let Vbe the subspace of R4consisting of all vectors v=

ï£®

ï£¯

ï£¯

ï£°

x

y

z

w

ï£¹

ï£º

ï£º

ï£»

for which

x+y+z=w. Find an orthonormal basis for V. (b) Let v1=ï£®

ï£°

1

2

3

ï£¹

ï£»,v2=ï£®

ï£°

1

âˆ’2

3

ï£¹

ï£»,v3=ï£®

ï£°

2

0

6

ï£¹

ï£»

be vectors in R3. Explain why the Gramm-Schmidt procedure applied to (v1, v2, v3)does not give

you 3 orthonormal vectors.

Problem 5. (a) A matrix Ais called symmetric if A=AT. Find all 2Ã—2symmetric matrices

which are also orthogonal. What about 3Ã—3? (b) Give an example of a 3Ã—3matrix which is

not orthogonal, but which is invertible and compute its QR factorization. (c) For which scalars

c1, c2, c3is the matrix

ï£®

ï£°

1 2 3

4c1c2

5 6 c3

ï£¹

ï£»

orthogonal?

1

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