This

**preview**shows half of the first page. to view the full**2 pages of the document.**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

.

.

.

ﬁnd the matrix Bfor the transformation Tin B-coordinates, and then ﬁnd 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 satisﬁes the following curious relation: Im(A) = ker(A).

Most of the time, Im(A) and ker(A) are subspaces living in totally diﬀerent 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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