# Algebra

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ALGEBRA AND ANALYTICAL GEOMETRY - MACROFACULTY, sem.II
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XI(B).
Vector Spaces - Basis
1. Which of the following are linear combination of u = (0, 1, 2) and v = (2, 0, -1) ?
(a) (2, 1, 3)
(b) (-2, 2, 5)
(c) (0, 0, 0)
2. Express the following as linear combinations of u1 = (1, 1, 1), u2 = (1, 0, 1), u3 = (0, -1, 1)
(a) (2, 0, 3)
(b) (3, 3, 2)
(c) (0, 0, 0)
1
-1
4
0
3. Which of the following are linear combination of A =
, B =
,
2
3
-2
-2
0
2
C =
?
1
4
0
0
6
-8
2
0
-1
7
(a)
(b)
(c)
(d)
0
0
-1
-8
5
10
8
5
4. Determine whether the given vectors span R4
(a) v1 = (1, 1, 1, 0), v2 = (1, 1, 0, 0), v3 = (1, 0, 0, 0), v4 = (1, 0, 0, -1)
(b) v1 = (1, 2, 3, 4), v2 = (-1, 1, 2, 0), v3 = (1, 1, -1, -1)
(c) v1 = (0, 1, -1, 1), v2 = (1, 1, 2, 3), v3 = (1, 1, 2, 2), v4 = (0, -1, 1, -2)
Remark: If rankA = dimV then set of vectors spans V (A - matrix with vectors in columns).
1
0
1
1
1
0
1
1
0
1
5. Let A1 =
, A
, A
, A
, A
.
1
1
2 =
0
1
3 =
1
0
4 =
0
0
5 =
0
1
Is S spanning set for M2x2 if
* S = {A1, A2, A3}
* S = {A3, A4, A5}
* S = {A1, A2, A3, A4, A5}
6. Let f (x) = sin2 x and g(x) = cos2 x. Which of the following lie in span{f, g}?
(a) cos 2x
(b) sin 2x
(c) 2
(d) 1 - cos 2x
(e) 3 - x2
(f) x + 2 sin2 x
(g) 0
7. Let p(x) = (1 - x)2. Determine whether p span(S), where
(a) S = {2 - x, 2x - x2, 6 - 5x + x2}
(b) S = {x2 - 1, x + 1}

(c) S = {x, 2, x2 3 - ln }
8. Let {u, v} be linearly independent set. Prove that the set {u + v, u - v} is linearly independent.
9. Determine whether S is a basis for the indicated vector space
(a) S = {(3, 2), (-1, 3)} for
2
R
(b) S = {(0, 1), (-1, 0), (1, 1)} for
2
R
(c) S = {(1, 5, 3), (0, 1, 2), (0, 0, 6)} for
3
R
(d) S = {(1, 1, 1, 1), (1, 1, 1, 0), (1, 1, 0, 0), (1, 0, 0, 0)} for
4
R
(e) S = {(0, 0, 0, 2), (0, 0, 1, 1), (0, 3, 3, 3), (4, 4, 4, 4), (1, -1, 2, 0)} for
4
R
(f) S = {1 + i, 1 - i} for C (what is a standard basis for C)
1

(g) S = {2, x + 1, -x2 - x - 1} for P2
10. Find the basis for R3 that includes vectors u = (1, 1, 1) and v = (1, 0, 1).
11. Find the basis for W = {(x, y, x - 2y) : x, y R}
12. Find the basis for W = {p(x) P2 : p(x) = a + bx2}
13. Find the basis for the set of all 4 x 4 diagonal matrices. What is the dimension of this vector space?
14. Find the basis for the set of all 4 x 4 symetric matrices. What is the dimension of this vector space?
15. Show that given sets are linearly independent
(a) S = {x, ex}
(b) S = {ex, e-2x, e3x}
(c) S = {1, x, xex}
(d) S = {ex sin 2x, ex cos 2x, 1}
(e) S = x, x + 2
(f) S = {sin x, cos x, tan x}
2