Welcome to an introduction
to matrix transformations.
Any function T that maps from R n to R m
is a transformation,
and we say T is a transformation from R n to R m.
Given an m by n matrix A,
the matrix transformation associated with matrix A
is T of vector x equals matrix A times vector x,
which is equal to sum output vector y.
So we can think of the matrix transformation
as a matrix function, or the inputs and outputs are vectors.
The input vectors are in R n,
and the output vectors are in R m,
where n is a number of columns in matrix A,
and m is a number of rows in matrix A.
Before we look at some examples,
let's define some vocabulary.
R n is the domain of T.
These are the input vectors with n components.
R m is the codomain of T.
For every vector x in R n,
the output vector T of vector x or vector y
is called the image of vector x under T.
And for every T of vector x or vector y in R m,
the input vector x is called the pre-image.
The set of all images T of x or vector y
is called the range of T.
The range of T is the column space of matrix A.
So looking at the diagram below, again, R n is the domain,
R m is the codomain, vector x is the input vector,
and the output vector is A times vector x or vector y.
We can say vector y is the image of vector x under T,
and vector x is the pre-image.
Also notice how the range may or may not be all of R m
since the range of T is the column space of matrix A,
the range can be a subset of R m.
Let's take a look at some examples.
We are given matrix A, which is a two by three matrix,
and we're given the matrix transformation
as T of vector x equals matrix A times vector x.
Since matrix A is a two by three matrix,
where m is two and n is three,
T's a transformation from R three to R two.
Next, for us to find the image of vector X,
which means we defined T of the vector two,
negative six, five.
To do this, we need to find the product of matrix A
and the given vector x.
I've already set this up below,
notice how the product is the vector 18, four.
If we look at the dimensions
of the product here on the left,
again, matrix A is a two by three matrix,
vector x is a three by one matrix,
because three equals three, we know we can multiply,
and the result is a two by one matrix,
which verifies this is a transformation
from R three to R two.
Let's take a look at this graphically.
The blue vector in R three is vector x, the input vector,
and the red vector, output vector
is the vector of the yellow x, y plane.
We can say the red output vector
is the image of the input vector under T,
and we can say the blue input vector is the pre-image.
Again, where our domain is R three
and the codomain is R two,
the x, y plane, which is in yellow.
And now let's take a look at a second example.
We're given matrix B.
Notice matrix B is a three by two matrix,
three rows and two columns.
And we're given the matrix transformation
as T of vector x equals matrix B times vector x.
Because matrix B is a three by two matrix,
T is a transformation from R two to R three.
We're asked to find the image of vector x,
which means we need to find T
of the vector negative three, four,
which I've already set up below.
Again, checking the dimensions of the product.
Because two equals two, we know we can multiply,
and the result is a three by one matrix,
verifying this is a transformation from R two to R three.
And the product is the vector negative three,
negative 10, seven.
Let's also look at this graphically.
This time have a transformation from R two to R three,
and therefore the input vector
is the blue vector in the x, y plane,
and the output vector is the red vector in R three.
The red output vector is called the image
of the blue input vector under T,
and the blue input vector is called the pre-image.
Before we go, let's talk about a linear transformation.
A function that maps one vector space
to another vector space
while preserving vector addition and scalar multiplication
is a linear transformation.
And there are two properties of linear transformations.
T of the sum of vectors u and v
is equal to T of vector u plus T of vector v,
and T of c times vector u is equal to c times T of vector u.
And since a matrix transformation
satisfies these two properties,
a matrix transformation is a linear transformation.
Let's go ahead and prove these two properties
for a matrix transformation.
First, we need to show that T of the sum of u and v
equals T of u plus T of v.
Well, T of the sum of u and v
is equal to matrix A times the sum of u and v,
so that on the left we have function notation,
on the right we have matrix A times the sum of u and v.
Well A times the sum of u and v
is equal to A times u plus A times v,
and A times u is T of u,
and A times v is T of v,
which is what we wanted to show.
T of the sum of u and v is equal to T of u plus the T of v.
Let's also prove T of c times u equals c times T of u.
Well T of c times u is equal to matrix A
times the product of c and u,
using the communicative and associative properties
of multiplication, we can write this
as c times the product of A and u,
and c times the product of A and u
does give us c times T of u.
I hope you found this introduction helpful.