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Assignment 3
Remark: About fortnightly assignment submission (Assignment 3), it should be given before October 29, 2017.
1. Check if the following vectors are linearly independent or not. If it is linear dependent, write a nontrivial relation of dependence. (a)
u1 =
1 1 1 1 1
, u2 =
1 2 3 4 5
, u3 =
1 −1 1 −1 1
.
(b)
v1 =
1 −1 1 −1
, v2 =
1 1 2 3
, v3 =
1 2 1 1
, v4 =
0 0 3 8
.
2. Let S be a linearly independent set. Let T be a nonempty subset of S. Show that T is also linearly independent.
3. Let A = [u1∼ , u2∼
, u3∼ ] where
u1∼ =
12
5
, u2∼ =
01
4
, u3∼ =
21
1
Determine if there exists J ∈ M3(R) such that JA = I.
4. Suppose v ∈ span〈v1, · · · , vk〉. Prove that span〈v1, · · · , vk, v〉 = span〈v1, · · · , vk〉.
5. The set of 2×2 symmetric matrices is a vector space with bases B = {[
1 0 0 0
] ,
[ 1 1 1 0
] ,
[ 1 1 1 1
]} and C =
{[ 1 1 1 0
] ,
[ 0 1 1 0
] ,
[ 0 1 1 1
]} as bases. Find the transition matrices from B to C
and from C to B.
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6. Let V be a vector space over R. Let 0∼ ∈ V be such that
u∼ + 0∼ = u∼ ∀u∼∈ V
Prove that
(a) 0 · u∼ = 0∼, ∀u∼∈ V
(b) −u∼ = (−1)u∼, ∀u∼∈ V
7. Determine if the set S = {
(x, y, z)T |xy = z2 }
is a subspace of R3 ?
8. Let
V = {
x1 x2 x3 x4
∈R4|x1 + x2 + x3 + x4 = 0}
By the definition of basis, show that
S =
1 0 0 −1
,
0 1 0 −1
,
0 0 1 −1
is a basis for V . What is dim V ?
9. Let Q be an invertible matrix with size n. Show that W = {Q−1AQ|A ∈ Mn(R)} is a subspace of Mn(R).
10. Suppose S is a five-dimensional subspace of R6. True or false?(give reasons.) (a) Every basis for S can be extended to a basis for R6 by adding one more vector. (b) Every basis for R6 can be reduced to a basis for S by removing one vector.
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