1 Vector spaces

1.0 Vectors and scalars

Scalar:

Examples:

Vector:

Examples:

Vector notation:

Norm of a vector:

Equality of two vectors:

1.1 Geometrical representation of vectors

1.2 Vector operations

Addition (parallelogram rule):

Scalar multiplication:

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1.2.1 Properties of vector addition

Commutativity:

Associativity:

Distributivity:

Zero vector (additive identity):

Negative vector (additive inverse):

1.2.2 Properties of scalar multiplication

Identity:

Distributivity:

Exercise 1: Given that A = 2B determine graphically A+ 2(BC).

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1.3 Angle and dot product

1.3.1 Angle between two vectors ~u and ~v

1.3.2 Dot product of two vectors ~u and ~v

Special case (i) = 2:

Special case (ii) ~u = ~v:

1.3.3 Projection of ~u on ~v

1.3.4 Application: Work done by a force ~F

Exercise 2: An object is pulled a distance of 100 m along a horizontal path by a constant force of25N. The force is applied at an angle of 30 above the horizontal. Find the work done by the force.

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1.4 The n-space

1.4.1 2-space

1.4.2 3-space

1.4.3 n-space

1.4.4 Properties in n-space

Exercise 3: Let ~u = (1, 3, 0,1), ~v = (2, 0, 1, 2), and ~w = (3, 5,2, 4). Find ~x if2~u ~x = 2~v + ~w 2~x.

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1.5 Dot product, norm, and angle in n-space

Suppose ~u = (u1, u2), ~v = (v1, v2) R2.

Exercise 4: Let ~u = (1, 0) and ~v = (2,2). Find , the angle between ~u and ~v.

Exercise 5: Let ~u = (2,2, 4,1) and ~v = (5, 9,1, 0). Find , the angle between ~u and ~v.

1.5.1 Properties of dot product

1.5.2 Properties of norm

Exercise 6: Expand (2~u ~v) (~w + 2~u).

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1.6 Schwarz inequality and triangle inequality

1.6.1 Schwarz inequality (or Cauchy-Schwarz inequality)

For any two vectors ~u and ~v, |~u ~v| ~u~v.

Proof. (Optional)

For any R, observe that (~u+ ~v) (~u+ ~v) 0. I.e.,

~u2 + 2(~u ~v) + 2~v2 0. (1)

Choose = ~u ~v~v2

and substitute in (1) to obtain

~u2 2(~u ~v)2

~v2+

(~u ~v)2

~v2 0. (2)

After simplifying (2) we get the desired inequality |~u ~v| ~u~v.

1.6.2 Triangle inequality

For any two vectors ~u and ~v, ~u+ ~v ~u+ ~v.

Proof. (Optional)

Consider

~u+ ~v2 = (~u+ ~v) (~u+ ~v)= ~u2 + 2(~u ~v) + ~v2

~u2 + 2|~u ~v|+ ~v2

~u2 + 2~u~v|+ ~v2 by the Schwarz inequality= (~u+ ~v)2

which gives the desired inequality ~u+ ~v ~u+ ~v.

Exercise 7: Let ~u = (1,2) and ~v = (0, 3). Verify Schwarz inequalty and triangle inequalitygraphically and algebrically.

Exercise 8: Let ~u = (1, 2, 4, 0) and ~v = (3,1, 2, 5). Verify Schwarz inequalty and triangle inequality.

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1.7 Orthogonality

1.7.1 Perpendicular vectors

1.7.2 Orthogonal vectors

1.7.3 Orthogonal set

Exercise 9: Let ~u1 = (2, 3,1, 0), ~u2 = (1, 2, 8, 3) and ~u3 = (9,6, 0, 1). Is { ~u1, ~u2, ~u3} an orthogonalset?

1.8 Normalization

1.8.1 Normal vector

Exercise 10: Normalize ~u = (2,3, 0, 1).

1.8.2 Orthonomal set

1.8.3 Kronecker delta function

Exercise 11: Let ~u1 = (1, 0, 0, 0), ~u2 = (0,12, 0, 1

2) and ~u2 = (0,

12, 0, 1

2). Is { ~u1, ~u2, ~u3} an

orthonomal set?

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1.9 Generalized vector spaces

Exercise 12: Show that Rn is a vector space.

Exercise 13: Let u : [0, 1] R be continuous. The sum of two functions u and v is given byu + v := u(x) + v(x) and scalar multiplication is given by u := u(x). Show that the functionspace S = {u(x)|u : [0, 1] R} is a vector space.

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1.9.1 Inner product (dot product)

1.9.2 Norm

1.10 Span

1.10.1 Linear combination of vectors

1.10.2 Span of vectors

Exercise 14: Find the span of {~u = (2, 3)}.

Exercise 15: Find the span of {~u = (2, 3), ~v = (4, 6)}.

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1.11 Subspaces

Exercise 16: Let ~u1 = (5, 1) and ~u2 = (1, 3). Is span{ ~u1, ~u2} = R2 or a subspace of R2?

1.12 Linear independence and dependence

Exercise 17: Let ~u1 = (1, 0), ~u2 = (1, 1) and ~u3 = (5, 4). Is { ~u1, ~u2, ~u3} linearly independent orlinearly dependent?

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Exercise 18: Let ~u1 = (1, 0), and ~u2 = (1, 1). Is { ~u1, ~u2} linearly independent or linearly dependent?

Exercise 19: Let ~u1 = (2,1), ~u2 = (0, 1) and ~u3 = (0, 0). Is { ~u1, ~u2, ~u3} linearly independent orlinearly dependent?

1.12.1 Theorem (Test for linear independence)

Exercise 20: Let ~u1 = (2, 0, 1,3), ~u2 = (0, 1, 1, 1) and ~u3 = (2, 2, 3, 0). Is { ~u1, ~u2, ~u3} linearlyindependent or linearly dependent?

Exercise 21: Let ~u1 = (1, 0, 1), ~u2 = (1, 1, 1), ~u3 = (1, 1, 2) and ~u3 = (1, 2, 1). Is { ~u1, ~u2, ~u3, ~u4}linearly independent or linearly dependent?

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1.12.2 Propositions

1.13 Basis

1.13.1 Theorem (Test for basis)

Exercise 22: Let ~e1 = (2, 1), and ~e2 = (2, 4). Is {~e1, ~e2} a basis for R?

Exercise 23: Expand ~u = (6, 2) in terms of the basis vectors ~e1 = (2, 1), and ~e2 = (2, 4).

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1.14 Dimension

1.14.1 Theorem (Test for dimension)

1.14.2 Orthogonal basis

Exercise 24: Expand ~u = (4, 3,3, 6) in terms of orthogonal basis vectors ~e1 = (1, 0, 2, 0), ~e2 =(0, 1, 0, 0), ~e3 = (2, 0, 1, 5), and ~e4 = (2, 0, 1,1) of R4.

1.14.3 Orthonormal basis

Exercise 25: Expand ~u = (4, 3,3, 6) in terms of orthonormal basis vectors in R4.

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