## Transcribed Text

Problem 1. Look at the following picture, Suppose you know that in the picture v1 and v2 are
orthogonal, e1 and e2 are orthogonal. (We will define what is orthogonal 2 weeks later. Now just use
your geometric intuition.)
1
Now suppose there is a mirror set on span(v2). We define T to be the reflection by span(v2)
(Remember span(v2) is the line that v2 stays)
(0) Is this a linear map or linear transformation?
(5) Find the matrix representation of T with respect to basis
e1 e2
(Using previous result,
do some substitution)
2
(6) Now take a look at that picture, draw −5e1, and find
where is T(
−5e1) in the picture. Write
down the coordinate of T(−5e1) with respect to basis e1 e2 . Now use your matrix representation in previous one to algebraically calculate what is T(−5e1). Then Compare.
(7) Look at the picture, tell me directly, what is the kernel of T? What is the dimension of
kernel?
(8) Look at the picture, tell me directly, what is the image of T?What is the dimension of image?
(1) Which of the following is T v1?
(A) v1 (B) − v1 (C) v2 (D) − v2 (E) 0
(2) Which of the following is T v2?
(A) v1 (B) − v1 (C) v2 (D) − v2 (E) 0
(3) Find
the m trix
represen ation of T with respect to basis
v1 v2
a (Write in the form
t
T v1 v2 = v1 v2 P)
(4) Find the matrix P by looking at the picture, such that
v1 v2
=
e1 e2
P, and with
your P,
and
do some
algebraic strategy on this equation to find the matrix Q, such that
e1 e2 = v1 v2 Q(Hint: what is the algebraic strategy? right multiply the inverse
of so
Problem 2. Suppose V, and W are two linear spaces over R, now assume
e1 e2 e3
be a
basis of V, and
1 2
be a basis of W. and
v1 v2 v3
be bunch of vectors in V, and
w1 w2
be bunch of vectors in W. and T is linear map from V to W Do the following question.
I’ll show an example. .
.
.
Suppose
v1 v2 v3
=
e1 e2 e3
2 1 4
5 2 7
6 1 1 ;
w1 w2
=
1 2
5 4
6 5
;
Te1 = 31 + 32
Te2 = 31 + 22
Te3 = 1 + 32
1 Represent T by the basis
e1 e2 e3
and
1 2
2 Determine whether
v1 v2 v3
and
w1 w2
are basis for domain and target respectively
3 if it is, represent linear transformation T with respect to
v1 v2 v3
and
w1 w2
1 Represent T by the basis
e1 e2 e3
and
1 2
We know directly,
T
e1 e2 e3
=
1 2
3 3 1
3 2 3
2 Determine whether
v1 v2 v3
and
w1 w2
are basis for domain and target respectively
We calculate
2 1 4
5 2 7
6 1 1
= −3 6= 0
5 4
6 5
= 1 6= 0
Thus the coordinate matrices are invertible, by the theorem, they are bases.
3 if it is, represent linear transformation T with respect to
v1 v2 v3
and
w1 w2
.
We want to express Tv1, Tv2, and Tv3 in terms of w1 and w2
.
Firstly let’s represent 1 and 2 in terms of w1 and w2
Because
w1 = 51 + 62
w2 = 41 + 52
From above, we express 1, 2 in terms of w1 and w2
1 = 5w1 − 4
Now we are ready:
Tv1 = T(2e1 + 5e2 + 6e3)
= 2Te1 + 5Te2 + 6Te3
= 2(31 + 32) + 5(31 + 22) + 6(1 + 32)
= 221 + 192
= 22(5w1 − 4w2) + 19(−6w1 + 5w2)
= −w1 + 8w2
And doing the same type of calculation, we get the expression of Tv2 and Tv3, to sum up we
have
Tv1 = −w1 + 8w2
Tv2 = 10w1 − 10w2
Tv3 = 54w1 − 59w2
So the matrix representation of T with basis
v1 v2 v3
and
w1 w2
is
T
v1 v2 v3
=
w1 w2
−1 10 54
8 −10 −59
With understanding of the method, do the exercises in the following pages.
6
(1)
v1 v2 v3
=
e1 e2 e3
1 1
1 1
1 ;
w1 w2
=
1 2
2 1
1 1
;
Te1 = 21 + 2
Te2 = 1
Te3 = 2
e
1 e2 e3
1 Represent linear map
T by the basis
2 Determine whether v1 v2 v3 and w1 w2
and12
are basis for domain and target respectively
3 if it is, represent linear transformation T with respect to
v1 v2 v3
and
Problem 3. How is everything going so far? Feel tons of calculation? Now we use matrix to redo
the 3rd part of the previous problem, look here.
v1 v2 v3
=
e1 e2 e3
2 1 4
5 2 7
6 1 1
w1 w2
=
1 2
5 4
6 5
;
Te1 = 31 + 32
Te2 = 31 + 22
Te3 = 1 + 32
Now we want to represent T with respect to bases
v1 v2 v3
and
w1 w2
. That is easy.
We know
T
e1 e2 e3
12
3 3 1
3 2 3
and
1 2
=
w1 w2
5 4
6 5 −1
We do the following steps
T
v1 v2 v3
= T
e1 e2 e3
2 1 4
5 2 7
6 1 1
=
1 2
3 3 1
3 2 3
2 1 4
5 2 7
6 1 1
=
w1 w2
5 4
6 5−1
3 3 1
3 2 3
2 1 4
5 2 7
6 1 1
=
w1 w2
−1 10 54
8 −10 −59
So that means the matrix representation of T with respect to
v1 v2 v3
and
w1 w2
is
T
v1 v2 v3
=
w1 w2
−1 10 54
8 −10 −59
Do the problem on the next page again usi
(1)
v1 v2 v3
=
e1 e2 e3
1 1 5
2 1
3 1
w1 w2
=
12
1 1
−1 1
;
Te1 = 41 + 22
Te2 = 21 + 22
Te3 = 22
represent linear transformation T with respect to
v1 v2 v3
and
w1 w2
Problem 4. Let V = M3×1(R) denote the space of 3 × 1 matrices over R, W = M2×1(R) denote
the space of 2 × 1 matrices over R, T is a linear map
T : V −→ W
And suppose we know:
T
1
2
5
1
2
; T
1
3
4
5
; T
−2
1
2
1
0
;
We want to realize T as a matrix multiplication.
(1) Show that {
1
2
5
4
1
3
−2
1
2
} is a basis for V (Write down the coordinate matrix
of this bunch of vectors with respect to natural basis. And try to show that coordinate matrix
is invertible.)
(2) Now with basis
1
2
5
4
1
3
−2
1
2
in the domain, and natural basis in the target. Write
down the matrix representation of linear transformation T.
13
e
(3) R lize linear tr a
ansformation
T as a matrix multiplication. (Hint, you have the form like
T e1 e2 e3 = 1 2 P, now because ei and i
are column
matrices, thus all the
thing except T could view as block matrices. Now treat e1 e2 e3 as a matrix, and
move it to the right. You get an expression of T equal to a matrix product. Then this
realizes T as a matrix )
(4) With your realization, calculate T
2
3
1
Problem 5. Suppose V = Px
2 = {ax2 + bx + c|a, b, c ∈ R}, W = Pt
2 = {at2 + bt + c|a, b, c ∈ R}.
Plug in t = x + 1 for polynomial in V gives an polynomial in W. Thus defines a linear map T. For
example
T(3x
2 + 2x + 3) = 3(t + 1)2 + 2(t + 1) + 3 = 3t
2 + 8t + 8
(1) Is this a linear map or linear transformation? (Hint: t,x is different letter)
(2) With basis
x
2 x 1
in domain, and basis
t
2
t
1
in target,
write down the
matrix
representation of linear map T.(present in the form T x
2 x 1 = t
2
t 1 P)
(3) Now
using your matrix representation, find the coordinate of
T(x
2 + 5x + 1) with respect to
basis t
2
t 1 (In this problem, only matrix multiplication is allowed)
(4) Find the kernel of this linear map.
(5) Show that T is an isomorphism
Problem 6. We want to find out all 2 × 2 matrices that can commute with 1
2
.
Suppose V = M2×2(R) the linear space of 2 × 2 matrices. Define a linear transformation:
T : V −→ V
T(M) = M
1
2
−
1
2
M
(1) With respect to the basis consisting of
1
1
,
1
−1
,
1
,
1
Find the matrix representation of this linear transformation.
(2) The matrices commuting with
1
forms a subspace in V. Can you find a basis of this 2
subspace? What is the dimension of the subspace? (Think how does this question related to
kernel of T)
(3) The matrices P, that can be represented as P = M
1
−
1
M for some M, 2 2
forms a subspace in V. Can you find a basis of the subspace? What is the dimension of the
subspace? (Think how does this question related to image
Problem 7. There are 3 brothers. Namely, A,B,C. They have some money originally. Suppose
there is neither income and costs. And they like to share their money. At the end of every month,
each of them will split his money into two equal part and give it to two others. For example. Suppose
A has 300 dollars at first month. At second month, A with give 150,150 to B,C respectively. and
B,C also do the same thing.
(0) Suppose this month, A has $8192, B has $4096, C has $8192. What amount will they have
after 1 year? (Please say: Give up)
(1) OK, don’t worry, suppose T is the linear transformation that transform the account status
this month to the status next month. With basis
e1 = A dollar that belongs to A
e2 = A dollar that belongs to B
e3 = A dollar that belongs to C
Write down the matrix representation of T with respect to
e1 e2 e3
(Hint, Te1 =
1
2
e2 +
1
2
e3, A dollar belongs to A this month would become 2 quarters belongs to B and 2
quarters belongs to C next month.)
(2) Now suppose
1 = e1 + e2 + e3
2 = e1 − e2
3 = e2 − e3
Show that
1 2 3
is a basis. And calculate T(1), T(2), T(3), represent your
result in terms of 1, 2, 3
20
(3) Calculate T
12(1), T
12(2), T
12(3), represent T
12 with respect to basis
1 2 3
(4) Now with the above calculation, represent T
12 with respect to basis
e1 e2 e3
(5) Suppose this month, A has $8192, B has $4096, C has $8192. What amount will they have
after 1 y

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