Introduction

Definition

A vector space is a set of objects with two operations: addition and multiplication by a scalar, which satisfy:

Checking

To verify that a set V is a vector space, one can:

1. Identify the set of objects

2. Identify the addition and multiplication by scalar operations in V

3. Verify axioms 1 and 6: closure under addition and closure under multiplication

4. Verify all other axioms

Examples

→ Set of real numbers ✓

→ Set of positive numbers ✗

→ Set of vectors of real numbers ✓

→ Set of nxm matrices ✓

Subspaces

Definition

A subset W of objects of a vector space V is called a subspace if it is also a vector space

Note that some properties are inherited by the subset

One needs only to verify axioms: 1, 4, 5, and 6

Theorem

If W is a subset of objects of a vector space V, W is also a vector space if and only if:

Axiom 1 holds for W

Axiom 6 also holds for W

Examples

→ A line through the origin in a plane ✓

→ A line that does not cross the origin in a plane ✗

→ A set of all points in a quadrant of a plane ✗

→ Set of diagonal matrices ✓

Span

Given a subset S of vectors in a vector space V as shown below:

\[ S = (\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}, \ldots, \mathbf{w}_{r}) \]

The set of all possible linear combinations of S is a subspace of V. We say that S span W

Linear Independence

Definition

Given a set of vectors S:

\[ S = (\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \ldots, \mathbf{v}_{r}) \]

S is said to be linearly independent if no vector in it can be expressed as a linear combination of the others

Theorem

A set of vectors S in a vector space is linearly independent if and only if:

\[ k_{1}\mathbf{v}_{1} + k_{2}\mathbf{v}_{2} + k_{3}\mathbf{v}_{3} + \ldots + k_{r}\mathbf{v}_{r} = \mathbf{0} \]

Only has the trivial solution

Checking for Linear Independence

Given a set of n vectors for an n-dimensional vector space, one can check if the set is linearly independent:

1. Write the set as a matrix

2. Check if its determinant is nonzero

Cordinates and Basis

Coordinates

Coordinate systems can be used to map points into ordered sets of numbers

Examples:

→ Rectangular coordinate systems in 2D

→ Polar coordinate systems in 2D

Frequently, unit perpendicular vectors are used to represent coordinate systems

Basis

If S is a set of vectors in a finite-dimensional vector space V, it is called a basis for V if:

(a) S spans V

(b) All vectors in S are linearly independent

Examples

Show that both sets of vectors below can form a basis for the 2D space:

\[ \mathbf{u}_{1} = (1, 0),\ \mathbf{u}_{2} = (0,1) \]

\[ \mathbf{u}_{1} = (1, 0),\ \mathbf{u}_{2} = (1,1) \]

Coordinates and Basis

If S is a basis for a vector space V, any vector in V can only be represented in exactly one way:

\[ \mathbf{v} = c_{1}\mathbf{v}_{1} + c_{2}\mathbf{v}_{2} + \ldots + c_{n}\mathbf{v}_{n} \]

The set of scalars c_i is the coordinate vector of v relative to S

\[ (\mathbf{v})_{S} = (c_{1}, c_{2}, \ldots,c_{n}) \]

Examples

Find the coordinates of the point (3, 4), in the following basis:

\[ \mathbf{u}_{1} = (1, 0),\ \mathbf{u}_{2} = (0,1) \]

\[ \mathbf{u}_{1} = (1, 0),\ \mathbf{u}_{2} = (1,1) \]

\[ \mathbf{u}_{1} = (1, 1),\ \mathbf{u}_{2} = (1,-1) \]

Change of Basis

Introduction

Sometimes a change of basis is necessary

Given a set of coordinates in a basis, how can one calculate the set of coordinates in a different basis?

If v is a vector in a space vector V, how are the coordinates [v]_B and [v]_B' related? B is the old basis and B' the new basis.

Transition Matrix

The solution is to create a matrix P with the following property:

Each column of P represents the coordinate vectors of the new basis relative to the old basis

This matrix P is called a transition matrix

P maps coordinates in a new basis back to its old basis

Example

\[ \mathbf{u}_{1} = (1, 0),\ \mathbf{u}_{2} = (0,1) \]

\[ \mathbf{u'}_{1} = (1, 0),\ \mathbf{u'}_{2} = (1,1) \]

Properties

Any invertible matrix can be shown to be a transition matrix from a basis formed by its columns to the standard basis

If P is the transition basis from B' to B (new basis to old), P^-1 is the transition matrix from B to B' (old to new)

Procedure

A simple way to find P is:

\[ [ \text{new basis} | \text{old basis} ] \xrightarrow{\text{row operations}} [ I | P^{-1} ] \]