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Let x,y, and z be vectors in a vector space W and . W has the following properties.

closure

(c1)

(c2)

Addition

(a1)

(a2)

(a3) and for all x

(a4) If then and

Multiplication

(m1)

(m2)

(m3)

(m4) for all

is a vector space.

What is a Vector?

The answer to this questions depends on who you ask. There are several ways to vectors.

(Physicist-geometric) Anything that has been assigned a direction and length.

(Computer Scientist-numeric) An ordered list of numbers.

(Mathematician-algebraic) An element in a vector space.

A vector space is a set of “vectors” that satisfy the 2 closure, 4 addition and 4 multiplication properties given to the right.

The set of column matrices we called is a vector space due to the properties of adding and scalar multiplication of matrices.

(a1)

(a4)

(a3)

(c2)

(m1)

(m4)

Algebra and Vectors

The properties of a vector space are the fundamental concepts needed in order to do basic algebraic manipulations like solving equations. The example to the right show how the properties apply to solving

Subspaces

A subset W of vectors may or may not form a vector space. A subset W of that is itself a vector space is called a subspace of . Any subset will satisfy the inherited properties (a1), (a2), (m1), (m2), (m3), (m4).

The theorem to the right shows exactly when a subset is a subspace.

- Subspace Theorem
- A subset W of is a subspace of if and only if W satisfies the following 3 properties:
- (a3) and for all x
- (c1) If then
- (c2) If and then

Show is not a subspace.

The vector since

Showing W is not a subspace

To show W is not a subspace you need to give a specific example of how W does not satisfy one of the properties.

Showing W is a subspace.

To show a subset W is a subspace you need to show that W satisfies all 3 conditions of the subspaces theorem.

Example

Show is a subspace.

1. Show (a3): which means

2. Show (c1): If with and then and

now

then

which means

3. Show (c2): If and with then

now

then which means

Show that is or is not a subspace.

This is not subspace. If and then , but

Example

Show that is or is not a subspace.

This is subspace.

1. Show (a3): which means

2. Show (c1): If with and

then

3. Show (c2): If and with

then

Let A be a matrix show that is a subspace of . (We will call this the kernel of matrix A.)

1. Show (a3): which means

2. Show (c1): If then and

then which means

3. Show (c2): If and then , then

which means

Example

Let v be a vector in , show that is a subspace of . (We call this the orthogonal space to the vector v.)

1. Show (a3): which means

2. Show (c1): If then and ,

then

3. Show (c2): If and then then

which means

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