School

University of Waterloo
Department

Mathematics

Course Code

MATH136

Professor

Patrick Roh

Lecture

2

Recall:

If

for any

then

is closed under addition. (Note: This is property 1 of

).

If

for any

and then is closed under scalar multiplication. (Property 6)

Definition

If a non-empty subset of satisfies the ten properties of then it is a subspace of .

Theorem: (Subspace Test)

Let be a non-empty subset of .

If

and

for all

and

, then

is a subspace of

.

Proof:

Properties 2, 3, 7, 8, 9, 10 must hold because we are applying the same operations as in .

Properties 1 and 6 are what is being checked.

Properties 4 holds since

for any

.

Properties 5 holds since

for any

Note: Our proof shows that any set that does NOT contain the zero vector is NOT a subspace.

Eg.

is NOT a subspace because

.

is a plane through the point

Eg.

Note:

so is non-empty.

Let

Let .

=0

Note:

Property 4 holds true.

Lec 4 - Subspace & Dot Product

Wednesday, January 11, 2012

9:26 AM

MATH 136 Page 1

Note:

By the subspace test, is a subspace of .

Note:

a line through the origin with direction vector

)

Eg.

Note:

so

is non-empty.

Let

But

is NOT a subspace.

Dot Product

Definition

Given vectors

the dot product of

and

is

The dot product is also referred to as the standard inner product or the scalar product (when

in

).

Note: The result of the dot product is a scalar, NOT a vector.

Eg

MATH 136 Page 2

Theorem (Properties of Dot Product)

Let

. Let

and

1.

2.

3.

Proof:

Let

1.

for

2.

Q.E.D.

3.

Definition

For

, the length or norm of

is defined as

.

Eg.

Definition

is a unit vector if

.

Theorem: (Property of Norms)

Let

and

MATH 136 Page 3

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