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Lecture 2

# Week 2 (lec 4-6) Lecture Notes

Department
Mathematics
Course Code
MATH136
Professor
Patrick Roh
Lecture
2

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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
.
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

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


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

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

















Theorem (Properties of Dot Product)
Let
. Let 
and

1.
2.
3.
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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