MATH 311 Section 501
Quiz on Linear Functions
Span e x , 1, e x , e 2x
1. Consider the vector space V
Spring 2013 P. Yasskin
with the usual addition and scalar multiplication
of functions. Two bases are:
f1
e x,
f2
1,
f3
ex,
f4
e 2x
and
F1
a. Find
e x,
F2
e
x
1,
F3
ex,
1
F4
ex
e 2x
C , the change of basis matrix from the F-basis to the f-basis.
f F
b. Find
C , the change of basis matrix from the f-basis to the F-basis.
F f
2. Consider the vector space W
Span 1, e x , e
x
with the usual addition and scalar multiplication of
functions. Two bases are:
e1
1,
e2
ex,
e3
e
x
and
E1
a. Find
1,
E2
sinh x
ex
e
2
x
E3
cosh x
ex
e
x
2
C , the change of basis matrix from the E-basis to the e-basis.
e E
b. Find
C , the change of basis matrix from the e-basis to the E-basis.
E e
1
3. With V and W as defined in #1 and #2, consider the function L : V
dp
2p
dx
a. First make sure the function L is well defined.
In other words, for p ae x b ce x de 2x V, verify that L p
W given by
Lp
W.
b. Show L is linear.
c. Find the Ker L . Give a basis. What is dim Ker L ?
Remember: The basis vectors must be functions in V.
d. Find the Im L . Give a basis. What is dim Im L ?
Remember: The basis vectors must be functions in W.
e. Let q
f. Find
4e
x
2e x
2e 2x and compute L q
directly from the definition of L.
A , the matrix of the linear map L from to the f-basis on V to the e-basis on W.
e f
g. Compute
h. Find
q
f
and recompute L q
L 4e
x
2e x
2e 2x
using
A.
e f
B , the matrix of the linear map L from to the F-basis on V to the E-basis on W.
E F
HINT:
Use
A , and two of
e f
i. Use
C
to compute
F f
j. Recompute L q
C,
e E
C,
E e
C
and/or
f F
q F , the components of q
L 4e
x
2e x
2e 2x
using
C.
F f
q
F
4e
x
and
2e x
2e 2x relative to the F-basis.
B .
E F
Then check your work by substituting sinh x and cosh x into the answer.
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