The Chapter 4 Super Summary
Use this as a quick reference guide if you’re in a hurry. But to really understand the material, or
for helpful examples (and nonexamples), please read the chapter summaries below it!
Formatting note
: writing math in a google document is kind of hard to deal with sometimes.
When I write something like
, I’m calling
a threerow vector with vertical entries
0,1,2.
Chapter 4.1.
●
A set
is a
vector space
if, for any
and
in
, and any scalar
,
○
is in
, and
○
is in
●
The
span
of vectors
is the space of all linear combinations of these vectors, i.e.
the vectors of the form:
where
are scalars.
Chapter 4.2.
●
A
linear transformation
is a function
from one vector space
to another
such
that, for any
and any scalar
,
○
○
●
Every linear transformation from
to
corresponds to an
matrix, and vice
versa.
●
A linear transformation
has four key related spaces:
○
Its
domain
○
Its
codomain
○
Its
range
, the subspace of
that’s the collection of all possible outputs
of
○
Its
kernel
, the subspace of
that’s the collection of all inputs that
maps to the zero vector
●
A linear transformation is
onetoone
if its kernel is just the zero vector. It’s
onto
if its
range is the same as its codomain. It’s
invertible
if it’s both.
●
The
column space
of a matrix
is the span of its column vectors, and the
null space
of
is the space of all vectors
such that
, i.e. the space of all solutions to the
homogeneous system.
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●
If
is a linear transformation and
is its corresponding
vector, then
the range of
is the column space of
and the kernel of
is the null space of
.
Chapter 4.3.
●
A collection of vectors
is
linearly dependent
if there exist scalars
, that
aren’t
all
zero such that:
A collection that’s not linearly dependent is called linearly independent.
●
For any matrix
, the columns of
form a linearly independent collection of vectors if
and only if the only solution to
is the trivial
.
●
A
basis
of a vector space
is a collection of vectors
such that:
○
is linearly independent
○
●
Be sure to check my summary of this chapter for instructions on how to find bases for
the null space, column space and row space!
Chapter 4.4.
●
Say
is a basis of a vector space
. For any
, there is a unique
coordinate form
, where if
then:
●
If
is a basis of the space
, then the
changeofcoordinates
matrix
is the matrix with columns
. This matrix is useful because:
is always invertible.
Chapter 4.5.
●
Every basis of a vector space
has the same number of vectors, and that number is
called its
dimension
.
●
If you have a set
of
vectors in
, you can get some information them based
on
:
○
If
, then
is linearly dependent. It may or may not span
.
○
If
, then either
is a basis of
, or it’s neither linearly
independent nor does it span
.
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 Spring '03
 BUSS
 Math, Linear Algebra, Vector Space, Space, Scalars

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