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Final

MATH 33AH Lecture Notes - Lecture 9: Orthogonal Complement, A Aa, Row Echelon FormExam


Department
Mathematics
Course Code
MATH 33AH
Professor
All
Study Guide
Final

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PRACTICE PROBLEMS FOR MATH 33A MIDTERM 2 (the majority of these are questions I consid-
ered for the midterm, but couldn’t fit):
1. Consider the matrix
A=
1203
0012
0012
Find a basis Ifor Im(A). What is rank(A)?
Find a basis Kfor ker(A). What is nullity(A)?
Is the vector
y=
7
2
2
1
in ker(A)? If so, find its K-coordinates (
y)K.
Find an orthonormal basis Ufor ker(A), starting from your basis Kin the earlier part of the
problem.
Find the matrix for orthogonal projection onto Im(A) in R4.
Find AATand ATA.
Show that the vector
b=
1
2
3
is not in Im(A). Therefore, the linear system A
x=
bis
inconsistent.
2. If Vis a vector subspace with orthogonal complement V, and
xis in V, what is the orthogonal
projection of
xonto V?
3. If I have a basis B={
v1,
v2,
v3,· · · ,
vn}for Rn, and I have a transformation Tthat sends each
basis element to the sum of the previous ones:
T(
v1) =
0
T(
v2) =
v1
T(
v3) =
v1+
v2
T(
v4) =
v1+
v2+
v3
.
.
.
nd the matrix Bfor the transformation Tin B-coordinates, and then nd B2, and B3. Is Bkthe
zero matrix for some exponent k? If so, what is the lowest such kthat accomplishes this?
4. If Ais an orthogonal matrix, and {
u1,
u2,
u3}is an orthonormal set of vectors, what is (A(
u1+
u2)) ·(A(
u1+
u3))?
5. If Vis a vector subspace in Rnwith orthogonal complement V, and I take a basis for Vtogether
with a basis for V, how many total vectors do I have in my collection? Must this resulting collection
be linearly independent? Must it be orthonormal?
6. Suppose we have an nxnmatrix Athat satises the following curious relation: Im(A) = ker(A).
Most of the time, Im(A) and ker(A) are subspaces living in totally dierent overall spaces. Why
is that not an issue here?
What can you say about the parity (odd or even) of n, and what are rank(A) and nullity(A) in
terms of n?
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