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Linear Algebra for Team-Based Inquiry Learning PREVIEW Edition

Steven Clontz, Drew Lewis

Section 3.1 Linear Transformations (AT1)

Subsection 3.1.1 Warm Up

Activity 3.1.1.

(a)
What is our definition for a set \(S\) of vectors to be linearly independent?
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(b)
What specific calculation would you perform to test is a set \(S\) of Euclidean vectors is linearly independent?
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Activity 3.1.2.

(a)
What is our definition for a set \(S\) of vectors in \(\IR^n\) to span \(\IR^n\) ?
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(b)
What specific calculation would you perform to test is a set \(S\) of Euclidean vectors spans all of \(\IR^n\) ?
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Subsection 3.1.2 Class Activities

Definition 3.1.3.

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
  1. \(T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})\) for any \(\vec{v},\vec{w} \in V\text{,}\) and
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  2. \(T(c\vec{v}) = cT(\vec{v})\) for any \(c \in \IR,\) and \(\vec{v} \in V\text{.}\)
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In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.
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Definition 3.1.4.

Given a linear transformation \(T:V\to W\text{,}\) \(V\) is called the domain of \(T\) and \(W\) is called the co-domain of \(T\text{.}\)
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described in detail following the image
The domain \(\IR^3\) is represented on the left by the \(xyz\) coordinate axes, along with an arbitrary vector \(\vec{v}\text{.}\) A curved dotted arrow to the right points to the co-domain, \(\IR^2\text{,}\) represented by the \(xy\) coordinate axes, along with an arbitrary vector labeled \(T(\vec{v})\text{.}\)
Figure 22. A linear transformation with a domain of \(\IR^3\) and a co-domain of \(\IR^2\)
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Observation 3.1.5.

One example of a linear transformation \(\IR^3\to\IR^2\) is the projection of three-dimensional data onto a two-dimensional screen, as is necessary for computer animation in film or video games.
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described in detail following the image
A computer generated image of a three dimensional teapot sitting in front of a screen that shows a flattened, two dimensional image of the same teapot. Several parallel black arrows point from identifiable points on the three dimensional teapot (such as the spout and handle) to the corresponding places on the two dimensional image.
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Figure 23. A projection of a \(3D\) teapot onto a \(2D\) screen
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Activity 3.1.6.

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*}
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(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*}
  1. \(\displaystyle \left[\begin{array}{c} x-u+z-w \\ 3y-3v \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} x+u-z-w \\ 3y+3v \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} x+u \\ 3y+3v \\ z+w \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} x-u \\ 3y-3v \\ z-w \end{array}\right]\)
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(b)
Compute the result of adding vectors after a \(T\) transformation:
\begin{equation*} T\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) + T\left( \left[\begin{array}{c} u \\ v \\ w \end{array}\right] \right) = \left[\begin{array}{c} x-z \\ 3y \end{array}\right] + \left[\begin{array}{c} u-w \\ 3v \end{array}\right] \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} x-u+z-w \\ 3y-3v \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} x+u-z-w \\ 3y+3v \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} x+u \\ 3y+3v \\ z+w \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} x-u \\ 3y-3v \\ z-w \end{array}\right]\)
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(d)
Compute the result of scalar multiplication before a \(T\) transformation:
\begin{equation*} T\left(c\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = T\left(\left[\begin{array}{c} cx \\ cy \\ cz \end{array}\right] \right) \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} cx-cz\\ 3cy \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} cx+cz \\ -3cy \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} x+c \\ 3y+c \\ z+c \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} x-c \\ 3y-c \\ z-c \end{array}\right]\)
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(e)
Compute the result of scalar multiplication after a \(T\) transformation:
\begin{equation*} cT\left(\left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = c\left[\begin{array}{c} x-z \\ 3y \end{array}\right] \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} cx-cz\\ 3cy \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} cx+cz \\ -3cy \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} x+c \\ 3y+c \\ z+c \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} x-c \\ 3y-c \\ z-c \end{array}\right]\)
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Activity 3.1.7.

Let \(S : \IR^2 \rightarrow \IR^4\) be given by
\begin{equation*} S\left(\left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x+y \\ x^2 \\ y+3 \\ y-2^x \end{array}\right] \end{equation*}
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(a)
Compute
\begin{equation*} S\left( \left[\begin{array}{c} 0 \\ 1 \end{array}\right] + \left[\begin{array}{c} 2 \\ 3 \end{array}\right] \right) = S\left( \left[\begin{array}{c} 2 \\ 4 \end{array}\right] \right) \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} 6 \\ 4 \\ 7 \\ 0 \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 5 \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} -3 \\ -1 \\ 7 \\ 5 \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} 6 \\ 4 \\ 10 \\ -1 \end{array}\right]\)
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(b)
Compute
\begin{equation*} S\left( \left[\begin{array}{c} 0 \\ 1 \end{array}\right] \right) + S\left( \left[\begin{array}{c} 2 \\ 3 \end{array}\right] \right) = \left[\begin{array}{c} 0+1 \\ 0^2 \\ 1+3 \\ 1-2^0 \end{array}\right] + \left[\begin{array}{c} 2+3 \\ 2^2 \\ 3+3 \\ 3-2^2 \end{array}\right] \end{equation*}
  1. \(\displaystyle \left[\begin{array}{c} 6 \\ 4 \\ 7 \\ 0 \end{array}\right]\)
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  2. \(\displaystyle \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 5 \end{array}\right]\)
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  3. \(\displaystyle \left[\begin{array}{c} -3 \\ -1 \\ 7 \\ 5 \end{array}\right]\)
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  4. \(\displaystyle \left[\begin{array}{c} 6 \\ 4 \\ 10 \\ -1 \end{array}\right]\)
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Activity 3.1.8.

Fill in the \(\unknown\)s, assuming \(T:\mathbb R^3\to\mathbb R^3\) is linear:
\begin{equation*} T\left(\left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right]\right) = T\left(\unknown \left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]\right) = \unknown T\left(\left[\begin{array}{c} 1 \\ 1 \\ 1 \end{array}\right]\right) = \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \end{array}\right] \end{equation*}
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Remark 3.1.9.

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 \(T(\vec v+\vec w)\not=T(\vec v)+T(\vec w)\text{.}\)
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  • Find specific values for \(\vec v\in V\) and \(c\in \IR\) such that \(T(c\vec v)\not=cT(\vec v)\text{.}\)
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  • Show \(T(\vec 0)\not=\vec 0\text{.}\)
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If you cannot do any of these, then \(T\) can be proven to be a linear transformation by doing both of the following:
  1. For all \(\vec v,\vec w\in V\) (not just specific values), \(T(\vec v+\vec w)=T(\vec v)+T(\vec w)\text{.}\)
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  2. For all \(\vec v\in V\) and \(c\in \IR\) (not just specific values), \(T(c\vec v)=cT(\vec v)\text{.}\)
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(Note the similarities between this process and showing that a subset of a vector space is or is not a subspace: Remark 2.3.14.)
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Activity 3.1.10.

(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?
  1. \(P\) is linear, but \(Q\) is not.
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  2. \(Q\) is linear, but \(P\) is not.
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  3. Both maps are linear.
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  4. Neither map is linear.
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(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.
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(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.
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Subsection 3.1.3 Individual Practice

Activity 3.1.11.

Let \(f(x)=x^3-1\text{.}\) Then, \(f\colon\IR\to\IR\) is a function with domain and codomain equal to \(\IR\text{.}\) Is \(f(x)\) is a linear transformation?
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Activity 3.1.12.

Consider two vectors \(\vec{u},\vec{v}\) and their sum \(\vec{u}+\vec{v}\text{.}\)
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(a)
Is it the case that rotating \(\vec{u}+\vec{v}\) about the origin by \(\frac{\pi}{2}=90^\circ\) is the same as first rotating each of \(\vec{u},\vec{v}\) and then adding them together?
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(b)
Is it the case that rotating \(5\vec{u}\) about the origin by \(\frac{\pi}{2}=90^\circ\) is the same as first rotating \(\vec{u}\) by \(\frac{\pi}{2}=90^\circ\) and then scaling by \(5\text{?}\)
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(c)
Based on this, do you suspect that the transformation \(R\colon\IR^2\to\IR^2\) given by rotating vectors about the origin through an angle of \(\frac{\pi}{2}=90^\circ\) is linear? Do you think there is anything special about the angle \(\frac{\pi}{2}=90^\circ\text{?}\)
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Activity 3.1.13.

In Activity 2.2.1, we made an analogy between vectors and linear combinations with ingredients and recipes. Let us think of cooking as a transformation of ingredients. In this analogy, would it be appropriate for us to consider "cooking" to be a linear transformation or not? Describe your reasoning.
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Subsection 3.1.4 Videos

Figure 24. Video: Showing a transformation is linear
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Figure 25. Video: Showing a transformation is not linear
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Subsection 3.1.5 Exercises

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Subsection 3.1.6 Mathematical Writing Explorations

Exploration 3.1.14.

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.
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Exploration 3.1.15.

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 \(f(\vec{v}^{-1}) = [f(\vec{v})]^{-1}\text{.}\)
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Subsection 3.1.7 Sample Problem and Solution

Sample problem Example B.1.12.
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