Math 110: Tutorial Problems — Week 13
1. Which of the following transformations are linear? Justify your claim.
x
x+y
x
xy
x
2x − 3y
T1
=
, T2
=
, T3
=
,
y
y+1
y
y
y
2x − 3y
T4
x
y
=
x3
x2
.
2. Let H be the plane in R3 defined by x + y + z = 0. Find the standard matrix of the
projection map PH : R3 → R3 onto H. What is the matrix of the map SH : R3 → R3
which reflects each vector in H?
x
x + 2y
x
x+y
3. Let T
=
and S
=
.
y
x+y
y
x+y
(a) Find T ◦ S and S ◦ T .
(b) Is T invertible? Is S invertible? For each map which is invertible, find its inverse.
Math 110: Tutorial Problems — Week 14
1. In the following determine whether the set W is a subspace of the vector space V .



a


3


−a
1. V = R , W =
|a∈R


2a
a b
2. V = M2,2 the vector space of 2×2 matrices with entries in R, W =
| ad ≥ bc .
c d
2. Is M2,2 spanned by
1 1
0 1
0 1
1 0
0 −1
,
,
and
?
1 0
1 1
1 0
3. Prove that every vector space has a unique zero vector.
Math 110: Tutorial Problems — Week 15
1 0
1 2
1 2
1 −2
1. Let u1 =
, u2 =
, u3 =
, u4 =
∈ M2 (R). Find
1 0
1 0
4 2
1 0
the E-coordinates of u1 , . . . , u4 and use these to determine whether S1 := {u1 , u2 , u3 }
and S2 = {u1 , u2 , u4 } are linearly independent. Also, find dim(span(Si )), for i = 1, 2.
2. Let ~v1 , . . . , ~vn , ~u ∈ V be vectors in a vector space V . Prove:
(a) span(~v1 , . . . , ~vn , ~u) = span(~v1 , . . . , ~vn ) if and only if ~u ∈ span(~v1 , . . . , ~vn ).
(b) If ~v1 , . . . , ~vn are linearly independent, then ~v1 , . . . , ~vn , ~u are linearly independent
if and only if ~u ∈
/ span(~v1 , . . . , ~vn ).
3. Let {~u, ~v , w}
~ be a linearly independent set of vectors in a vector space V , and let
S1 = {~u + ~v , ~v + w,
~ ~u + w}
~ and S2 = {~u − ~v , ~v − w,
~ ~u − w}.
~
(a) Is S1 linearly independent? (Justify!) Find dim(W1 ), where W1 = span(S1 ).
(b) Is S2 linearly independent? (Justify!) Find dim(W2 ), where W2 = span(S2 ).