# 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

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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