Math 369 HW #6
Due at the beginning of class on Friday, October 17
Reading: Sections ILT, SLT, CB, MR.
Problems:
1. For each of the following, determine whether the given function is a linear transformation.
(a) f : R3 → R2 given by
 
x
3x + 2y




y
f
=
5z
z
(b) g : R2 → R2 given by
x−y
x
=
g
2xy
y
(c) h : P2 (R)[z] → R3 given by

c−b
h(a + bz + cz 2 ) = b + a
4a

2. (a) Does there exist a linear transformation T : R2 → R3 such that
 
 
 
1
1
−1
2
0
1
= 5?
and
T
= 3 ,
T
= −2 ,
T
3
1
0
2
1
2
Explain your answer.
(b) Let g : P(R)[z] → P(R)[z] be a linear transformation with
2
g(1)
=P
x2(R
+)[1,z] be ag(x)
= xtransformation
− 1,
and such
g(xthat
) = x2 + x + 1.
(b) Let g : P (R )[
z] !
linear
2
)=
What is g(3x2 g−(12x
− x5)?+ 2;
What is g(2x2 5x + 1)?
3. Consider the following figure:
3. Consider the following figure.
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g( x2 ) = x2 + x + 1.
3;
g( x ) = x
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???.
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4..
(1)
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(a) Sketch the change to the Figure (1) after applying the linear transformation
✓ ◆ 

p
p
x
cos
sin
x
4
4
T
=1
.
p
p
y
sin 4 cos 4 y
(b) Describe a linear transformation T which when applied to Figure (1) would produce the
following image.
. . .??.??
??. . 6. . . . . . .
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?????
??.? ?. ? .
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???.
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4..
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(a) Sketch the figure which results
after applying
the to
following
linear
to the
(a) Sketch
the change
the Figure
(1) transformation
after applying the
linear transformat
✓
◆


above figure:
cos p4
sin p4
x
x
cos π/4 sin π/4 T x x
=
.
p
p
R
=
.
y
sin
cos
y
4
4
y
− sin π/4 cos π/4
y
(b) Describe a linear transformation T which when applied to Figure (1) wo
(b) Describe the linear transformation S which, when applied to the above figure, produces:
following image.
?. . 6. . . . . . .
. . .??.??
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?????
??.???. ? . .
??. ???.??????.// .
?.? . ?. ?/ .
?. . ?/.
. ?
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6
4. Suppose W and V are vector spaces and that f 2 L(W, V ) is a linear tran
4. Suppose V and W are vector spaces
and that
T :V →
nullspace
(or kernel)
of W
f is is a linear transformation. Prove
that null(T ) is a subspace of V .
null( f ) = {w 2 W : f (w) = 0}.
5. Suppose V and W are vector spaces and that F : V → W is a linear transformation.
Prove that null( f ) is a subspace of W.
(a) Suppose F (v1 ) = F (v2 ) 5.
forSuppose
some v1 ,W
v2 and
∈ V .VShow
that vspaces
null(F
1 − v2 ∈
are vector
and
that).f 2 L(W, V ) is a linear trans
(b) On the other hand, suppose
) for some u1 , u2 ∈ V . Show that
poseuw
, wu222∈Wnull(F
are vectors.
1 1−
F (u1 ) = F (u2 ).
(a) Suppose f (w1 ) = f (w2 ). Show that w1 w2 2 null( f ).
(b) Suppose w1 w2 2 null( f ). Show that f (w1 ) = f (w2 ).
Challenge: Recall from calculus that derivatives are linear transformations from
C • [0, 1] (here C • [0, 1] is the set of continuous functions on [0, 1] that have all
d
“smooth”.) Use a geometric argument to describe the null space of dx
as consta
f ( x ) = c. Then apply the result of problem (5) to conclude that two functions
derivative if, and only if, they differ by a constant.
Achter/Wilson
Colorado State University
2
36