Quick quiz of some essential vocabulary
Sums of subspaces.
Here are some questions to test your intuition about sums:
Fact
Proof
Here are some questions to test your intuition about sums:
Direct sums.
Direct sums.
This is the extra condition here.
Direct

Eigenvalues, eigenvectors, and
abstract linear operators.
The key fact we use here
The key fact we use here
Because that diagram commutes.
Because v is an eigenvector of T.
Because is a linear transformation.
This equals sign means that the vectors
on the

Bases for vector spaces
Passing comment.
Whats my point here?
Passing comment.
Choose a basis then
take
co-ordinate vectors.
A good co-ordinate system should do two things
1. It should assign one co-ordinate vector to every
point in a logical well-defined

Matrix representatives of linear transformations
of a vector space V to itself.
By the important theorem
about compositions.
Fact
Using the change-of-coordinates matrix to change the
chosen reference bases for a matrix representation of a
linear transform

Using bases to understand
linear transformations.
Linear transformations and bases.
Linear transformations and bases.
A picture of this
A picture of this
?
?
?
?
An alternative Illustration of this construction.

Isomorphisms: our first look.
Graph A:
Graph B.
Isomorphisms of vector spaces.
Isomorphism
Isomorphic
Isomorphism
Isomorphic
The previous fact (that two isomorphic vector spaces have the same dimension) is
exactly what we expected, according to the princi

Linear Transformations
Most specific topics in abstract mathematics can be viewed (at first glance) as
the study of two things.
Vector spaces.
Linear Transformations.
A set of vectors V and a field F with:
1. A way to add vectors v,w2 V:
v+w2 V.
2. A way

Decomposing vector spaces into subspaces:
Sums and direct sums.
Sums of subspaces.
Here are some questions to test your intuition about sums:
Fact
Proof
Direct sums.
Direct sums.
Direct sums.

The linear transformation determined by a matrix
The linear transformation determined by a matrix
The domain of the matrix A
Subspaces associated to linear transformations
Every linear transformation has two important subspaces associated with it.
m kg
We

Some comments on C([0,1]).
Some comments on C([0,1]).
+
(-1.5)
=
=
Some comments on C([0,1]).
+
Question
Answer
The key thing to notice about
these functions is that any pair
will be non-zero on different
sets of points.
Comment for the future
Comment for

The relationships between the ideas of
subspaces and dimension.
Subspaces and dimension
Subspaces and dimension
Summary of our discussion about subspaces and dimension
Fact #1:
Fact #2:
Fact #3:
Sums, direct sums and bases.
From direct sums to bases
and b

Today well think about 3 key facts about bases.
So why does this algorithm work?
So why does this algorithm work?
So why does this algorithm work?
So why does this algorithm work?
Key Fact #2.
Finite dimensional V.
Linearly
Independent S.
Basis B.
Key fac

Doing geometry in abstract vector spaces:
Inner product spaces
2,3
v
w
1.
Vectors have length.
2.
We can measure angles.
5
,1
2
A
2
y
1
2 1
x
Question
Question
Question
The key to doing geometry in an abstract vector
space is the existence of an inner

The norm in an inner product space: length.
Property I:
Take careful note:
The modulus of a number
is represented by one
vertical bar: |c|.
But the norm of a vector is
represented by two vertical
bars: v.
Property I:
Property II:
Property III:
Property I:

Orthogonal decomposition
Another look at reflections.
Another look at reflections.
Another look at reflections.
Reflections.
This means we should choose w=(-b,a).

Recall that a vector space over a field F:
Axiom I:
Axiom II:
Axiom III:
Axiom IV:
Axiom V:
Axiom VI:
Axiom VII:
Axiom VIII:
Rows almost always go
in the first position.
E.g.
Columns go in the
second position.
Of course, we could work
with different field

Diagonalization The important case of matrices.
Diagonalization and similarity of matrices.
So why is it useful to know that a matrix is similar
to a diagonal matrix?
Matrices which are diagonal are often very easy to use.
For example, they are very easy

Some preliminary comments about numbers
The rational
number
system.
The real
number
system.
The complex
number
system.
Some preliminary comments about numbers
Some preliminary comments about numbers
Linear algebra lives in two worlds
Rn
Matrices
Matrix
al

Summary of some key points from the last lecture
Sums of subspaces.
Here are some questions to test your intuition about sums:
Fact
Proof
Here are some questions to test your intuition about sums:
Direct sums.
Direct sums.
This is the extra condition here

How do we decode these questions?
a and b are real numbers.
i stands for the
square root of -1.
Now lets go back and decode these questions:
NO!
Now lets go back and decode these questions:
YES!
Now lets go back and decode these questions:
YES!

Eigenvalues, eigenvectors, and
the characteristic polynomial.
A
v v .
The key fact about eigenvalues:
if and
only if
The key fact is now:
if and only if
if and only if
if and only if
We can use this equation
to determine eigenvalues.
They are

Invertible functions, Isomorphisms, and so on.
Invertible linear
transformations.
Linear
transformations
which are both
1-1 and onto.
?
Isomorphisms.
The main result we are aiming for:
Invertible linear
transformations.
Linear
transformations
which are bo

Orthogonal decompositions.
This theorem feels a lot like some kind of direct-sum decomposition.
Can we interpret it as an example of a direct-sum decomposition?
To interpret the previous decomposition theorem as an example of a direct-sum
decomposition, w

Where do Orthonormal bases come from?
Where do Orthonormal bases come from?
Gram-Schmidt
orthogonalization
procedure.
Gram-Schmidt orthogonalization
Step 1.
After i steps:
Gram-Schmidt orthogonalization
Why does Gram-Schmidt orthogonalization work?
After

MATRIX REPRESENTATIONS OF LINEAR TRANSFORMATIONS.
Recall that linear algebra lives in two worlds
Abstract vector
spaces
Abstract linear
transformations
Rn
Matrices
Matrix
algorithms
In this world:
In this world:
Easier to prove things.
Easier to see the

Recall that a vector space over a field F:
Axiom I:
Axiom II: Axiom III:
Axiom IV: Axiom V: Axiom VI: Axiom VII: Axiom VIII:
Rows almost always go in the first position.
Columns go in the second position.
Of course, we could work with different fields.
E.

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Some preliminary comments about numbers
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Some preliminary comments about numbers
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