Subsection A Digression Into Functions
In class, we have already seen this definition, but we state it here for reference:
Definition 3.21.
Let \(A\) and \(B\) be sets. A function \(f\) from \(A\) to \(B\) is a way of associating to each element of \(A\) a unique element of \(B\text{.}\) We use the notation \(f\colon A\to B\) to denote that \(f\) is a function from \(A\) to \(B\text{.}\) For \(a\in A\text{,}\) we denote that \(f\) associates \(a\) with \(\in B\) as \(f(a)=b\) or as \(a\stackrel{f}{\mapsto} b\text{.}\) We call \(A\) the domain of \(f\) and \(B\) the codomain of \(f\text{.}\) The image of \(f\) is
\begin{equation*}
\mathrm{im}(f) = \{b\in B\colon \text{ there exists }a\in
A\text{ such that }f(a)=b\}\text{.}
\end{equation*}
Sometimes the image is called the range.
We now introduce two properties of functions that will be important in this course and your further study of mathematics.
A function \(f: C \rightarrow D\) is one-to-one if \(f(x)=f(y)\) implies \(x=y\text{.}\) This means that each input gets sent to a different output by the function \(f\text{.}\) Alternately, you can say a on-to-one function does not map two different inputs to the same output. For functions that we can graph in the \(xy\)-plane, being one-to-one is the same as passing the horizontal line test. A one-to-one function is also called an injection or said to be injective.
Example 3.22.
Any linear function from \(\mathbb{R}\) to \(\R\) is one-to-one. Suppose \(f(x)=ax+b\) with \(a\neq 0\text{.}\) To prove that \(f\) is one-to-one, we assume that \(f(x)=f(y)\) and show that \(x=y\text{:}\)
\begin{align*}
f(x) \amp = f(y)\\
ax+b\amp = ay+b\\
ax\amp =ay\\
x\amp =y
\end{align*}
Note that we used \(a\neq 0\) to divide both sides by \(a\text{.}\)
The domain is important when considering if a function is one-to-one. To see why, consider the function \(f\colon \R\to
\R\) with rule \(f(x)=x^2\text{.}\) Since \(f(-1)=1=f(1)\) but \(-1\neq 1\text{,}\) we see that this function is not one-to-one. However, if we change the domain and consider \(g\colon \R^+\to\R\) defined on the positive real numbers \(\R^+\) given by the rule \(g(x)=x^2\text{,}\) we see that \(g\) is one-to-one.
A function \(f:C \rightarrow D\) is onto if for all \(d\in D\text{,}\) there exists \(c\in C\) such that \(f(c)=d\text{.}\) In other words, a map \(f\) is onto if the image of \(f\) is all of \(D\text{.}\) An onto function is also said to be surjective or a surjection.
Example 3.23.
To prove that a function is onto, we take an arbitary element of the codomain and find an element of the domain that the function maps to the element of the codomain. We generally do this by first doing calculations on scratch paper and then demonstrating that the element of the domain “works” as our proof. For instance, consider the linear function \(f(x)=ax+b\) from \(\R\) to \(\R\) with \(a\neq
0\text{.}\) This function is onto. To prove this, let \(d\in\R\text{.}\) We will show that \(f\left(\frac{d-b}{a}\right)
=d\text{:}\)
\begin{align*}
f\left(\frac{d-b}{a}\right) \amp =
a\left(\frac{d-b}{a}\right) +b \\
\amp= (d-b)+b\\
\amp = d\text{.}
\end{align*}
Since we have started with an arbitrary lement \(d\) of the codomain and found an element of the domain that is mapped to \(d\text{,}\) we have proved that \(f\) is onto.
The codomain is essential to deciding if a function is onto. For instance, the function \(x\mapsto x^2\) (with domain \(\R\)) is not onto if the codomain is \(\R\text{,}\) as there is no real number that squares to \(-1\text{.}\) However, if we change the codomain to be all nonnegative real numbers, then the function is onto as it maps \(\sqrt{d}\) do \(d\) for all nonnegative real numbers \(d\text{.}\)
Question 3.39.
For each of the functions from \(\mathbb{R}\) to \(\mathbb{R}\) below state whether the function is either 1-1 but not onto, onto but not 1-1, 1-1 and onto, or not 1-1 and not onto.
(a)
\(f(x) =e^x\)
(b)
\(f(x) =x^2(1-x)\)
(c)
\(f(x) =\sin(x)\)
(d)
\(f(x) =x^3\)
For any property above that a function is missing, can you change the domain and/or codomain so that the function does have the property?
Subsection Functions on vector spaces
Linear transformations are the “nice” functions from a vector space to a vector space. In particular, linear transformations preserve the operations of scalar multiplication and vector addition.
Definition 3.24.
A function \(T\) from a vector space \(V\) to a vector space \(W\) is a linear transformation if for every \(\vec{v_1},\vec{v_2} \in V\) and \(c \in \mathbb{R}\)
Question 3.40.
Prove that the map \(T: \mathbb{R}^n
\rightarrow \mathbb{R}^m\) given by \(T(\vec{x}) = A\vec{x}\) is linear, where \(A\) is an \(m\) by \(n\) real-valued matrix.
Eventually we will be able to state a lot of linear transformations as a matrix transformation like in the problem above, but we will not be able to do this in general.
Question 3.41.
Prove that the map \(T: \mathbb{P} \to \mathbb{P}\) given by \(T(f)=\dfrac{df}{dt}\) is linear. You may use your calculus knowledge.
Question 3.42.
For each of the following functions, determine if the function is a linear transformation. Remember to justify your reasoning and answers.
(a)
\(f_1:\mathbb{P} \to \mathbb{R}\) where \(f_1(\vec{p})=\) the degree of the polynomial \(\vec{p}\)
(b)
\(f_2:\mathbb{P} \to \mathbb{R}\) where \(f_2(\vec{p})= \vec{p}(t=1)\)
(c)
\(f_3:\mathbb{R}^2 \to \mathbb{R}^3\) where \(f_3(\colvec{a\\ b})=\colvec{a+b\\ a-b\\ b+1}\)
(d)
\(f_4:\mathbb{R}^3 \to \mathbb{R}^2\) where \(f_4(\colvec{a\\ b\\ c})=\colvec{a+b\\ a-c}\)
(e)
\(f_5:\mathbb{R}^3 \to \mathbb{R}^2\) where \(f_5(\colvec{a\\ b\\ c})=\colvec{2\\ a+b\\ c^2}\)
Theorem 3.25.
If \(T\colon V\to W\) is a linear transformation and a set of vectors \(\{v_1,v_2,v_3\}\) is linearly dependent, then the set \(\{T(v_1),T(v_2),T(v_3)\}\) is linearly dependent.
Question 3.43.
Give a counterexample to the following statement: If \(T\colon V\to W\) is a linear transformation and a set of vectors \(\{v_1,v_2,v_3\}\) is linearly independent, then the set \(\{T(v_1),T(v_2),T(v_3)\}\) is linearly independent.
Theorem 3.26.
If \(T\) is a linear transformation from \(V\) to \(W\text{,}\) then \(T(\vec{0}_V)=\vec{0}_W\text{.}\)
Question 3.44.
If a linear transformation, \(T\text{,}\) sends the vector \(\vec{e_1}=\colvec{1\\ 0}\) to \(\colvec{3\\ -1\\ 1}\) and sends the vector \(\vec{e_2}=\colvec{0\\ 1}\) to \(\colvec{1\\ 0\\ 2}\text{,}\) find the following:
(a)
\(T\left(\colvec{3\\ 0}\right)\)
(b)
\(T\left(\colvec{0\\ 5}\right)\)
(c)
\(T\left(\colvec{a\\ b}\right)\)
Question 3.45.
Find a matrix \(A\) such that for the transformation in the previous problem \(T(\vec{x})=A\vec{x}\text{.}\)
Definition 3.27.
If \(T\) is a linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\text{,}\) then the standard matrix presentation of \(T\) is a \(m\) by \(n\) matrix
\begin{equation*}
A=[T(\vec{e_1}) \quad T(\vec{e_2}) \quad ... \quad T(\vec{e_n}) ]
\end{equation*}
where \(\vec{e_i}\) is the \(i\)-th elementary vector of \(\R^n\text{.}\) Note that \((\vec{e_i})_j =
\delta_{i,j}\text{,}\) where \(\delta\) is the Dirac delta function defined by
\begin{equation*}
\delta_{i,j}=\left\{ \begin{array}{cc} 0 \amp \mbox{if
}i\neq j\\ 1 \amp \mbox{if } i = j \end{array} \right.
\end{equation*}
The vector \(\vec{e_i}\) can also be thought of as the \(i\)-th column of \(Id_n\text{,}\) the \(n\) by \(n\) identity matrix. Because of how we defined the standard matrix presentation, only transformations from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) will have standard matrix presentations. In particular, the standard matrix presentation keeps track of where the standard basis vectors (\(\vec{e_i}\)) go under the transformation \(T\text{.}\)
Exercise 3.28.
Write out \(\vec{e_1}\text{,}\) \(\vec{e_2}\text{,}\) and \(\vec{e_3}\) from \(\mathbb{R}^3\text{.}\) What is the result of multiplying \(\begin{bmatrix} a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i
\end{bmatrix}\) by \(\vec{e_1}\text{?}\) What about \(\vec{e_2}\text{?}\) \(\vec{e_3}\text{?}\)
What would this mean for the following matrix product:
\begin{equation*}
\begin{bmatrix} a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i \end{bmatrix} \begin{bmatrix} \vec{e_1}\amp \vec{e_2}\amp \vec{e_3} \end{bmatrix}
\end{equation*}
Question 3.46.
Determine the standard matrix presentation \(A\) for the following \(T\text{:}\)
(a)
\(T: \mathbb{R}^2 \to \mathbb{R}^2\) reflects points over the vertical axis
(b)
\(T: \mathbb{R}^2 \to \mathbb{R}^2\) rotates points clockwise by \(\pi/2\)
(c)
\(T: \mathbb{R}^2 \to \mathbb{R}^2\) rotates points by \(\pi\) and then flips points over the vertical axis
Exercise 3.29.
If a linear transformation, \(T\text{,}\) sends the vector \(\colvec{1\\ 1}\) to \(\colvec{-2\\ 2}\) and sends the vector \(\colvec{-1\\ 1}\) to \(\colvec{0\\ 2}\text{,}\) find the following:
(a)
\(T\left(\colvec{1\\
0}\right)\)
Hint.
How can you write \(\colvec{1\\
0}\) as a linear combination of \(\colvec{1\\ 1}\) and \(\colvec{-1\\ 1}\text{?}\)
(b)
\(T\left(\colvec{0\\ 1}\right)\)
(c)
\(T\left(\colvec{a\\ b}\right)\)
(d)
Find the standard matrix presentation for \(T\)
Theorem 3.30.
Let \(T_{\vec{0}}\) be the function from \(V\) to \(W\) such that \(T(\vec{x})=\vec{0}_W\) for every \(\vec{x} \in V\text{.}\) Let \(Id_V\) be the identity map on \(V\text{,}\) \(Id_V(\vec{x}) = \vec{x}\) for every \(\vec{x} \in V\text{.}\)
The function \(T_{\vec{0}}\) is a linear transformation.
The function \(Id_V\) is a linear transformation.
The null space, or kernel, of a linear transformation \(T:V \rightarrow W\) is the set of inputs that get mapped to the zero vector of \(W\text{.}\) That is \(Null(T)=\{\vec{x}\in V \mid T(\vec{x}) = \vec{0_W}\}\text{.}\)
Question 3.47.
Is \(\vec{b}=\colvec{0\\ 2\\ 1}\) in the image of the linear transformation \(T(\vec{x})=A\vec{x}\) where \(A= \begin{bmatrix} 1\amp 2 \\ 3 \amp 4\\0\amp 0 \end{bmatrix}\text{?}\) Justify your answer without doing any matrix operations.
Hint.
Write the corresponding matrix equation as a vector equation.
Exercise 3.31.
Let \(A=\begin{bmatrix} 1\amp 2\amp 3
\\4\amp 5\amp 6 \end{bmatrix}\text{.}\) Find the image and null space of \(T\) where \(T(\vec{x}) =A \vec{x}\text{.}\) Remember to state the image and null space so that the reader can most easily verify whether a vector is in the set or not.
Question 3.48.
Let \(T\) from \(\mathbb{R}^2\) to \(\mathbb{P}_2\) be given by \(T \left( \colvec{a\\ b} \right) = a +(a+b)t+(a-b)t^2\text{.}\)
(a)
Prove \(T\) is linear.
(b)
Compute the image of \(T\text{.}\)
(c)
Compute the null space of \(T\text{.}\)
Question 3.49.
Let \(V\) be the vector space of polynomials in \(x\) and \(y\text{.}\)
(a)
Show the transformation \(T\) that maps \(f\) to \(\dfrac{\partial f}{\partial x}\) is a linear transformation.
(b)
Compute the null space of \(T\text{.}\)
(c)
Compute the range of \(T\text{.}\)
Theorem 3.32.
If \(T\) is a linear transformation from \(V\) to \(W\text{,}\) then \(null(T)\) is a subspace of \(V\text{.}\)
Theorem 3.33.
If \(T\) is a linear transformation from \(V\) to \(W\text{,}\) then \(\mathrm{im}(T)\) is a subspace of \(W\text{.}\)
Question 3.50.
Let \(T\) from \(\mathbb{R}^2\) to \(\mathbb{P}_2\) be given by \(T \left( \colvec{a\\ b} \right) = a +(a+b)t+(a-b)t^2\text{.}\)
(a)
Is \(T\) one-to-one?
(b)
Is \(T\) onto?
Question 3.51.
Give an example of a linear transformation from \(\mathbb{R}^2\) to \(\mathbb{R}^3\) that is one-to-one.
Question 3.52.
Give an example of a linear transformation from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) that is onto.
Question 3.53.
Give an example of a linear transformation from \(\mathbb{R}^3\) to \(\mathbb{R}^2\) that is onto.
Question 3.54.
If the set of columns of a \(m\) by \(n\) matrix \(A\) are linearly independent, does the set of columns of \(A\) span all of \(\mathbb{R}^m\text{?}\)
Question 3.55.
If the set of columns of a \(m\) by \(n\) matrix \(A\) are linearly independent, is the image of \(T(\vec{x})=A\vec{x}\) all of \(\mathbb{R}^m\text{?}\)
Theorem 3.34.
A linear transformation \(T:V \rightarrow
W\) is onto if and only if \(\mathrm{im}(T)=W\text{.}\)
Theorem 3.35.
For each linear transformation \(T\) from \(V\) to \(W\text{,}\) \(null(T)=\{ \vec{0} \}\) if and only if \(T\) is one-to-one.
