Determine if a map between Euclidean vector spaces is linear or not.
Subsection3.1.1Class Activities
Definition3.1.1.
A linear transformation (also called a linear map) is a map between vector spaces that preserves the vector space operations. More precisely, if \(V\) and \(W\) are vector spaces, a map \(T:V\rightarrow W\) is called a linear transformation if
\(T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})\) for any \(\vec{v},\vec{w} \in V\text{,}\) and
\(T(c\vec{v}) = cT(\vec{v})\) for any \(c \in \IR,\) and \(\vec{v} \in V\text{.}\)
In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.
Definition3.1.2.
Given a linear transformation \(T:V\to W\text{,}\)\(V\) is called the domain of \(T\) and \(W\) is called the co-domain of \(T\text{.}\)
Figure19.A linear transformation with a domain of \(\IR^3\) and a co-domain of \(\IR^2\)
Observation3.1.3.
One example of a linear transformation \(\IR^3\to\IR^2\) is the projection of three-dimesional data onto a two-dimensional screen, as is necessary for computer animiation in film or video games.
Figure20.A projection of a \(3D\) teapot onto a \(2D\) screen
Activity3.1.4.
Let \(T : \IR^3 \rightarrow \IR^2\) be given by
\begin{equation*}
T\left(\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right)
=
\left[\begin{array}{c} x-z \\ 3y \end{array}\right].
\end{equation*}
(a)
Compute the result of adding vectors before a \(T\) transformation:
\begin{equation*}
T\left(
\left[\begin{array}{c} x \\ y \\ z \end{array}\right] +
\left[\begin{array}{c} u \\ v \\ w \end{array}\right]
\right)
=
T\left(
\left[\begin{array}{c} x+u \\ y+v \\ z+w \end{array}\right]
\right)
\end{equation*}
In summary, any one of the following is enough to prove that \(T:V\to W\) is not a linear transformation:
Find specific values for \(\vec v,\vec w\in V\) such that \(T(\vec v+\vec w)\not=T(\vec v)+T(\vec w)\text{.}\)
Find specific values for \(\vec v\in V\) and \(c\in \IR\) such that \(T(c\vec v)\not=cT(\vec v)\text{.}\)
Show \(T(\vec 0)\not=\vec 0\text{.}\)
If you cannot do any of these, then \(T\) can be proven to be a linear transformation by doing both of the following:
For all \(\vec v,\vec w\in V\) (not just specific values), \(T(\vec v+\vec w)=T(\vec v)+T(\vec w)\text{.}\)
For all \(\vec v\in V\) and \(c\in \IR\) (not just specific values), \(T(c\vec v)=cT(\vec v)\text{.}\)
(Note the similarities between this process and showing that a subset of a vector space is or is not a subspace: Remark 2.3.7.)
Activity3.1.8.
(a)
Consider the following maps of Euclidean vectors \(P:\mathbb R^3\rightarrow\mathbb R^3\) and \(Q:\mathbb R^3\rightarrow\mathbb R^3\) defined by
\begin{equation*}
P\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right)=
\left[\begin{array}{c} -2 \, x - 3 \, y - 3 \, z \\ 3 \, x + 4 \, y + 4 \, z \\ 3 \, x + 4 \, y + 5 \, z \end{array}\right]
\hspace{1em} \text{and} \hspace{1em} Q\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right)=
\left[\begin{array}{c} x - 4 \, y + 9 \, z \\ y - 2 \, z \\ 8 \, y^{2} - 3 \, x z \end{array}\right].
\end{equation*}
Which do you suspect?
\(P\) is linear, but \(Q\) is not.
\(Q\) is linear, but \(P\) is not.
Both maps are linear.
Neither map is linear.
(b)
Consider the following map of Euclidean vectors \(S:\mathbb R^2\rightarrow\mathbb R^2\)
\begin{equation*}
S\left( \left[\begin{array}{c} x \\ y \end{array}\right]\right)= \left[\begin{array}{c} x + 2 \, y \\ 9 \, x y \end{array}\right].
\end{equation*}
Prove that \(S\)is not a linear transformation.
(c)
Consider the following map of Euclidean vectors \(T:\mathbb R^2\rightarrow\mathbb R^2\)
\begin{equation*}
T\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right)= \left[\begin{array}{c} 8 \, x - 6 \, y \\ 6 \, x - 4 \, y \end{array}\right].
\end{equation*}
Prove that \(T\)is a linear transformation.
Subsection3.1.2Videos
Figure21.Video: Showing a transformation is linearFigure22.Video: Showing a transformation is not linear
If \(V,W\) are vectors spaces, with associated zero vectors \(\vec{0}_V\) and \(\vec{0}_W\text{,}\) and \(T:V \rightarrow W\) is a linear transformation, does \(T(\vec{0}_V) = \vec{0}_W\text{?}\) Prove this is true, or find a counterexample.
Exploration3.1.10.
Assume \(f: V \rightarrow W\) is a linear transformation between vector spaces. Let \(\vec{v} \in V\) with additive inverse \(\vec{v}^{-1}\text{.}\) Prove that \(f(\vec{v}^{-1}) = [f(\vec{v})]^{-1}\text{.}\)