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Lecture 2

Week 2 (lec 4-6) Lecture Notes

by OneClass31850 , Winter 2012
11 Pages
111 Views

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



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.
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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Description
Lec 4 SubspaceDot ProductWednesday January 11 2012926 AMRecall Iffor anythenis closed under addition Note This is property 1 of Iffor any andthenis closed under scalar multiplication Property 6Definition If a nonempty subsetofsatisfies the ten properties ofthen it is a subspace ofTheorem Subspace Test Letbe a nonempty subset of Ifandfor alland thenis a subspace ofProof Properties 2 3 7 8 9 10 must hold because we are applying the same operations as inProperties 1 and 6 are what is being checked Properties 4 holds sincefor any Properties 5 holds since for anyNote Our proof shows that any set that does NOT contain the zero vector is NOT a subspaceEgis NOT a subspace because is a plane through the point Eg Note sois nonemptyLetLet Note 0Property 4 holds true MATH 136 Page 1NoteBy the subspace testis a subspace ofNotea line through the origin with direction vectorEg Note sois nonemptyLet Butis NOT a subspaceDot ProductDefinition Given vectorsthe dot productof and is The dot product is also referred to as the standard inner productor the scalar productwhen inNote The result of the dot product is a scalar NOT a vectorEgMATH 136 Page 2
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