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Lessons for Linear Algebra: Play, Learn, Love Linear Algebra

Mark Fitch

Section 2.3 Transformations

Goals.

We will
  • identify a type of function that is important to linear algebra,
  • prove a couple, useful theorems about this type of function,
  • learn to test if functions have this property, and
  • discover a connection between matrix multiplication and these functions.

Subsection 2.3.1 Experiment: Property

A transformation is a function (synonym). The properties that define a linear transformation are illustrated in Activity 2.3.1.

Activity 2.3.1.

\begin{equation*} f( [x_1, x_2, x_3]^T) = [x_1+x_2,x_2+x_3,x_3+x_1]^T. \end{equation*}
(a)
Evaluate the following: \(f([1,3,-2]),\) \(f([1,2,5]),\) \(-2f([1,3,-2]),\) \(3f([1,2,5]),\) \(f([1,3,-2])+f([1,2,5])\text{.}\)
(b)
Evaluate \(f[-2[1,3,-2])\) and \(f(3[1,2,5])\text{.}\)
(c)
Compare these two results to the first set. Conjecture a property of this function.
Solution.
\(f(a\vec{x})=af(\vec{x})\text{.}\)
(d)
Evaluate \(f([1,3,-2]+[1,2,5])\text{.}\)
(e)
Compare this result to the first set. Conjecture a property of this function.
Solution.
\(f(\vec{x}+\vec{y})=f(\vec{x})+f(\vec{y})\text{.}\)
These two properties define a linear transformation.

Subsection 2.3.2 Linear Transformation Theorems

Definition 2.3.1. Linear Transformation.

A function \(T\) is a linear transformation if and only if
  1. \(T(a\vec{x})=aT(\vec{x}),\) and
  2. \(T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})\text{.}\)
First we explain why \(f(\vec{0})=\vec{0}\text{.}\) It relies on the first, defining property of linear transformations.

Proof.

\begin{align*} f(\vec{0}) & = f(0\vec{x})\\ & = 0f(\vec{x})\\ & = \vec{0}. \end{align*}
This theorem sometimes provides a quick way to show a transformation is not linear. Consider the following two examples.

Example 2.3.3.

Determine whether \(T([x_1,x_2,x_3]^T)=[x_1+1,x_2+1,x_3+1]^T\) is a linear transformation. Note \(T(\vec{0})=[0+1,0+1,0+1]=[1,1,1]\text{.}\) Because this is not \(\vec{0}\text{,}\) this transformation is not linear.

Example 2.3.4.

Determine whether \(T([x_1,x_2,x_3]^T)=[x_1^2,x_2^2,x_3^2]^T\) is a linear transformation. Note \(T(\vec{0})=\vec{0}\text{.}\) This is insufficient to determine whether the transformation is linear. (Spoiler alert) It is not linear.

Proof.

\begin{align*} f(a\vec{x}+b\vec{y}) & = f(a\vec{x})+f(b\vec{y}) \text{ by def. part 2}\\ & = a f(\vec{x})+b f(\vec{y}) \text{ by def. part 1}. \end{align*}
Note the right hand side of this theorem is both parts of the defintion of linear transformation. Thus we can use the left hand side to determine if a transformation is linear.

Example 2.3.6.

Consider the following transformation. \(j([x_1,x_2,x_3]^T) = [1,0,x_1+x_2+x_3]^T\text{.}\) For this we will use these two vectors we chose for no particular reason: \(\vec{u} = [1,2,1]^T\) and \(\vec{v} = [-3,-3,5]^T\text{.}\)
First we calculate
\begin{align*} j(\vec{u}) & = \left[ 1,0,4 \right]^T.\\ j(\vec{v}) & = \left[ 1,0,-1 \right]^T.\\ 5j(\vec{u})-2j(\vec{v}) & = \left[ 3,0,22 \right]^T.\\ j(5\vec{u}-2\vec{v}) & = \left[ 1,0,22 \right]^T.\\ j(5\vec{u}-2\vec{v}) & \ne 5j(\vec{u})-2j(\vec{v}). \end{align*}
Based on this comparison we can see that this transformation is not linear.

Subsection 2.3.3 The Convenience of Being Linear

The following activity illustrates how the nature (definition) of linear transformations provides a convenient way to work with them.

Activity 2.3.2.

Consider
\begin{equation*} f( [x_1, x_2, x_3]^T) = [x_1+x_2,x_2+x_3,x_3+x_1]^T. \end{equation*}
Note it is a linear transformation.
(a)
Evaluate \(f(\vec{v})\) at \(\vec{v}=[1,-2,3]\text{.}\)
(b)
Write \(\vec{v}\) as a linear combination of \(\vec{e}_1=[1,0,0]^T, \) \(\vec{e}_2=[0,1,0]^T, \) \(\vec{e}_3=[0,0,1]^T. \)
(c)
Evaluate \(f(\vec{e}_1),f(\vec{e}_2),f(\vec{e}_3)\text{.}\)
(d)
Evaluate \(f(\vec{v})\) using the linear combination above and the definition of linear transformation. Do not just plug it in (that is the same as the first way).

Subsection 2.3.4 Matrix Notation

The following activity illustrations another way to calculate linear transformations.

Activity 2.3.3.

Use the linear transformation
\begin{equation*} f( [x_1, x_2, x_3]^T) = [x_1+x_2,x_2+x_3,x_3+x_1]^T. \end{equation*}
(a)
Using \(f\) from above evaluate \(f([1,0,0]^T),f([0,1,0]^T),f([0,0,1]^T)\text{.}\)
(b)
Evaluate \(f([1,-2,3]^T)\text{.}\)
(c)
Evaluate \(\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] \left[\begin{array}{r} 1 \\ -2 \\ 3 \end{array}\right]\text{.}\)
(d)
Compare the results of the evaluation and the matrix multiplication.

Checkpoint 2.3.7.

Using the previous case as an example, find a matrix for the following linear transformations.
(a)
\(g([x_1,x_2,x_3]^T)=[x_3,x_1,x_2]^T\text{.}\)
(b)
\(f([x_1,x_2,x_3]^T)=[x_1+x_2,x_2+x_3]^T\text{.}\)

Subsection 2.3.5 Big Theorem: Linear Transformations

The following activity illustrates extending the big theorem to include properties of linear transformations.

Activity 2.3.4.

For this activity use
\begin{equation*} A=\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right], \;\; B=\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \end{array} \right] \end{equation*}
(a)
For what vectors, \(\vec{b},\) will \(A\vec{x}=\vec{b}\) have a solution?
(b)
What does this say about the range of \(f(\vec{x})=A\vec{x}\text{?}\)
(c)
What does this say about the transformation? Note transformations are functions but not always one-to-one or onto.
(d)
How many solutions are there to \(A\vec{x}=\vec{b}\) for a single vector \(\vec{b}\text{?}\)
(e)
What does this say about the transformation?

Activity 2.3.5.

For this activity use
\begin{equation*} A=\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right], \;\; B=\left[ \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 1 \end{array} \right] \end{equation*}
(a)
For what vectors, \(\vec{b},\) will \(B\vec{x}=\vec{b}\) have a solution?
(b)
What does this say about the range of \(g(\vec{x})=B\vec{x}\text{?}\)
(c)
What does this say about the transformation? Note transformations are functions but not always one-to-one or onto.
(d)
How many solutions are there to \(B\vec{x}=\vec{b}\) for a single vector \(\vec{b}\text{?}\)
(e)
What does this say about the transformation?
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