## Contents

### VECTOR SPACE

A **vector space** is a set of vectors that defines addition V x V --> V and scalar multiplcation cV --> V that satisfy the following properties:

1. **Communative Property**: u + v = v + u

2. **Associative Property**:

**·** Of addition: (u + v) + w = u = (v + w)

**·** Of multiplication: (ab)v = a(bv)

3. **Zero Property**: There exist some **0**∈V such that **0** + v = v

4. **Inverse Property**: For every v∈V there is some -v where v + -v = 0

5. **Identity Property**: 1v=v

6. **Distributive Property**: a(u + v) = au + av & (a + b)u = au + bu & c(du) = (cd)u

**Example**

Prove that vector addition and scalar multiplication define R^2 as a vector space.

$ \left[\begin{array}{cc}x1\\x2\end{array}\right] + \left[\begin{array}{cc}y1\\y2\end{array}\right] = \left[\begin{array}{cc}x1+y1\\x2+y2\end{array}\right] $

$ r*\left[\begin{array}{cc}x1\\x2\end{array}\right] = \left[\begin{array}{cc}rx1\\rx2\end{array}\right] $

First, define v1, v2, w1, w2, u1, u2 as elements in R^2 and a,b,c,d as scalars.

Second, check each of the properties in the defintion of a vector space. If a single property fails the entire proof fails.

1. **Communative Property**:

$ \left[\begin{array}{cc}v1\\v2\end{array}\right] + \left[\begin{array}{cc}w1\\w2\end{array}\right] = \left[\begin{array}{cc}v1+w1\\v2+w2\end{array}\right] $

2. **Associative Property**:

**·** Of addition:

$ \left[\begin{array}{cc}u1+v1\\u2+v2\end{array}\right] + \left[\begin{array}{cc}w1\\w2\end{array}\right] =\left[\begin{array}{cc}u1\\u2\end{array}\right] + \left[\begin{array}{cc}v1+w1\\v2+w2\end{array}\right] $

**·** Of multiplication:

$ (ab)*\left[\begin{array}{cc} v1\\v2\end{array}\right] = \left[\begin{array}{cc}(ab)v1\\(ab)v2\end{array}\right] = \left[\begin{array}{cc}a(bv1)\\a(bv1)\end{array}\right] = a*(b*\left[\begin{array}{cc}v1\\v2\end{array}\right] $

3. **Zero Property**:

$ \left[\begin{array}{cc}v1\\v2\end{array}\right] + \left[\begin{array}{cc}0\\0\end{array}\right] = \left[\begin{array}{cc}v1\\v2\end{array}\right] $

4. **Inverse Property**:

$ \left[\begin{array}{cc}v1\\v2\end{array}\right] + \left[\begin{array}{cc}-v1\\-v2\end{array}\right] = \left[\begin{array}{cc}0\\0\end{array}\right] $

5. **Identity Property**:

$ 1*\left[\begin{array}{cc}v1\\v2\end{array}\right] = \left[\begin{array}{cc}v1\\v2\end{array}\right] $

6. **Distributive Property**:

$ a*\left[\begin{array}{cc}u1+v1\\u2+v2\end{array}\right] = \left[\begin{array}{cc}au1\\au2\end{array}\right] + \left[\begin{array}{cc}av1\\av2\end{array}\right] $

&

$ (a + b)*\left[\begin{array}{cc}u1\\u2\end{array}\right] = a*\left[\begin{array}{cc}u1\\u2\end{array}\right] + b*\left[\begin{array}{cc}u1\\u2\end{array}\right] $

&

$ c*\left[\begin{array}{cc}du1\\du2\end{array}\right] = (cd)*\left[\begin{array}{cc}u1\\u2\end{array}\right] $

Because all of these are true you can conclude that vector addition and scalar multiplication indeed define R^2 as a vector space.

### SUBSPACE

A **subspace** is a subset of a vector space. To be a subspace of vectors the following must be true:

1. One set must be a **subset** of another set

2. The set must be closed under **scalar multiplication**

3. The set must be closed under **vector addition**

## Proving one set is a subset of another set

Given sets A and B we say that is is a subset of B if every element of A is also an element of B, that is,

x∈A implies x∈B

**Basic Outline of the Proof that A is a subset of B:**

**·** Suppose x ∈ A

1. Say what it means for x to be in A

2. Mathematical details

3. Conclude that x satisfies what it means to be in B

**·** Conclude x∈B

**Example**

Let A be the set of scalars divisible by 6 and let B be the even numbers. Prove that A is a subset of B.

**·** Suppose x ∈ A:

1. What it means for x to be in A: x = 6k for any scalar k

2. x = 2 × (3k)

3k = C

3. What it means for x to be in B: x = 2C

**·** Conclude x∈B

## Closed Under Scalar Multiplication

A set of vectors is closed under scalar multiplication if for every **v**∈V and every c∈\mathbb{R} we have c**v**∈V

**Basic Outline of the Proof V is Closed Under Scalar Multiplication:**

**·** Suppose **v**∈V and c∈\mathbb{R}

1. Say what it means for **v** to be in V

2. Mathematical details

3. Conclude that c**v** satisfies what it means to be in V

**·** Conclude c**v**∈V

## Closed Under Vector Addition

A set of vectors is closed under vector addition if for every **v** and **w** ∈ V we have **v** + **w** ∈ V

**Basic Outline of the Proof V is Closed Under Vector Addition:**

**·** Suppose **v** and **w** ∈ V

1. Say what it means for **v** and **w** to be in V

2. Mathematical details

3. Conclude that **v**+ **w** satisfies what it means to be in V

**·** Conclude **v** + **w** ∈ V

**Example**

Let V be the set of points in R^2 such that x=y

**·** Suppose **v** and **w** ∈ V

1. What it means for **v** and **w** to be in V :

**v** = (v1, v2) and v1 = v2

**w** = (w1, w2) and w1 = w2

2. **z** = **v** + **w** = (v1+w1, v2+w2) = (v1+w1, v1+w1)

3. What it means for **z** to be in V: v1+w1 = v2+w2

**·** Conclude **z** = **v** + **w** ∈ V

*Explanation of how to determine a subspace. Information referenced from Wabash College MA 223, Spring 2011*