Transcribed Text
Problem 1. Now, I choose a point O on your paper. Then your paper (if extend to infinity) become a
linear space, denote as V
(1) Suppose O was chosen in the following picture. Now v, w ∈ V , draw the vector that corresponds to this two element in the graph.
(2) Suppose our linear space is a right linear space and if we have u = v + 2w, correct the
historical mistake notation.(Hint: put the scalars on the right please), and then write u in
the form
v w
∗
∗
, after that, draw u in the picture:
Problem 2. Now suppose I choose O to be origin in the following picture, v1, v2, v3 as shown
(1) Draw
v1 v2 v3
−1
2
3
(2) Does
v1, v2, v3 in the picture linearly independent? If not, find a nonzero 3 × 1 matrix
∗
∗
∗
over R such that
v1 v2 v3
∗
∗
= 0, is this kind of matrix unique? If not,
write couple more.(the equation
v1 v2 v3
∗
∗
∗= 0 is called a linear relation of
v1 v2 v3
o
(3) Write 3 more distinct column matrix
∗
∗
∗such that
v1 v2 v3
∗
∗
∗
v1 v2 v3
−1
2
3
Problem 3. Suppose
e1 e2
is a basis for a linear space V over R. if
u = e1 + 2e2
v = e1 − e2
(1) Write
u v
as
e1 e2
multiplied by some matrix.
(2) Is
u v
linearly independent? prove it.
(3) Can
u v
spans the whole space? prove it.
(4) Is
u v
a basis? prove it.
(5) What is the coordinate of 4e1 − e2 with respect to the basis
u
Problem 4. Suppose V = M2×2(R), addition defined by matrix addition, multiplication defined by
usual scalar multiplication. Thus V become a linear space. Determine whether the following subsets
is linear subspace. Explain.
(1) {
a b
c d

a b
c d 1
2
=
0
0
}
(2) {
a b
c d

a b
c d 1
2
=
1
0
}
(3) {
a b
c d

a b
c d
is invertible}
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(4) {
a b
c d
rank
a b
c d
= 0}
(5) {
a b
c d

a b
c d2 1
1 2
=
2 1
1 2 a b
c d
}
Problem 5. Over R, Please choose an 2×2 matrix A that you like most, and with your A, complete
the following exercises.
(0) Write your A over here.
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(1) Calculate A2
(2) Is I, A, A2
linearly dependent? If it is, find a nontrivial linear relation. If not, check your
calculation in (1) and do this exercise again.(I is unit matrix. nontrivial linear relation
means a notallzero scalar linear combination to produce zero vector)
Problem 6. Let V be a linear space over F, suppose (e1, e2, · · · , en) and (1, 2, · · · , m) are both
basis for V .
(1) Does there exist unique matrix P such that (1, 2, · · · , m) = (e1, e2, · · · , en)P? Why? What
is the size of P?
(2) Does there exist unique matrix Q such that (e1, e2, · · · , en) = (1, 2, · · · , m)Q? Why?
What is the size of Q?
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(3) Prove that P Q = In, and QP = Im(Hint: Using equation in (2) to substitute something
in (1), and using equation in (1) to substitute something in (2), you would find something.
And by left cancellation rule of independent vectors, you get what you want)
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