Definition. A function \(T: V \to W\) from a subspace \(V\) of \(\mathbb R^n\) to a subspace \(W\) of \(\mathbb R^m\) is called a linear transformation if

1. \(T(\overrightarrow{u}+\overrightarrow{v})= T(\overrightarrow{u})+T(\overrightarrow{v})\) for all \(\overrightarrow{u}, \overrightarrow{v} \in V\) and


2. \(T(c\overrightarrow{v})=cT(\overrightarrow{v})\) for for all \(\overrightarrow{v}\in V\) and all scalars \(c \in \mathbb R\).

In short, a function \(T: V \to W\) is a linear transformation if it preserves the linearity among vectors:
\(T(c\overrightarrow{u}+d\overrightarrow{v})= cT(\overrightarrow{u})+dT(\overrightarrow{v})\) for all \(\overrightarrow{u}, \overrightarrow{v} \in V\) and all scalars \(c,d \in \mathbb R\).

Example.

1. The projection \(T:\mathbb R^3 \to \mathbb R^3\) of \(\mathbb R^3\) onto the \(xy\)-plane in \(\mathbb R^3\) is defined by \[T \left( \left[\begin{array}{c}x_1\\x_2\\x_3 \end{array} \right] \right) = \left[\begin{array}{c}x_1\\x_2\\0 \end{array} \right] \text{ for all } \overrightarrow{x}=\left[\begin{array}{c}x_1\\x_2\\x_3 \end{array} \right] \in \mathbb R^3.\] Sometimes it is simply denoted by \(T(x_1,x_2,x_3)=(x_1,x_2,0)\) in terms of row vectors. To show it is a linear transformation, let \(\overrightarrow{x}=(x_1,x_2,x_3)\) and \(\overrightarrow{y}=(y_1,y_2,y_3)\) in \(\mathbb R^3\) and \(c,d\in \mathbb R\). Then \[\begin{align*} T(c\overrightarrow{x}+d\overrightarrow{y}) &= T(cx_1+dy_1,cx_2+dy_2,cx_3+dy_3)\\ &= (cx_1+dy_1,cx_2+dy_2,0)\\ &= (cx_1,cx_2,0)+(dy_1,dy_2,0)\\ &=cT(\overrightarrow{x})+dT(\overrightarrow{y}). \end{align*}\]


2. For the matrix \(A=\left[\begin{array}{rr}1&2\\0&1 \end{array}\right]\), define the shear transformation \(T:\mathbb R^2\to \mathbb R^2\) by \(T(\overrightarrow{x})=A\overrightarrow{x}\). Let \(\overrightarrow{x}, \overrightarrow{y}\in \mathbb R^2\) and \(c,d\in \mathbb R\). Then \[T(c\overrightarrow{x}+d\overrightarrow{y})=A(c\overrightarrow{x}+d\overrightarrow{y}) = cA \overrightarrow{x}+dA \overrightarrow{y}= cT(\overrightarrow{x})+dT(\overrightarrow{y}).\] Thus \(T\) is a linear transformation which transforms the square formed by \((0,0)\),\((1,0)\),\((1,1)\),\((0,1)\) to the parallelogram formed by \((0,0),(1,0),(3,1),(2,1)\).

Definition. A matrix transformation is the linear transformation \(T:\mathbb R^n\to \mathbb R^m\) defined by \(T(\overrightarrow{x})=A\overrightarrow{x}\) for some \(m\times n\) matrix \(A\). It is denoted by \(\overrightarrow{x} \mapsto A\overrightarrow{x}\).

From the definition of a linear transformation we have the following properties.
Property For a linear transformation \(T: V \to W\) where \(V\leq \mathbb R^n\) and \(W\leq \mathbb R^m\),

1. \(T(\overrightarrow{0_n})=\overrightarrow{0_m}\) and


2. for all \(\overrightarrow{v_1}, \ldots,\overrightarrow{v_k} \in V\) and all \(c_1,\ldots,c_k \in \mathbb R\), \[T(c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+\cdots+c_k\overrightarrow{v_k}) = c_1 T(\overrightarrow{v_1})+c_2 T(\overrightarrow{v_2})+\cdots+c_k T(\overrightarrow{v_k}).\]

Example. Consider the function \(T:\mathbb R^3 \to \mathbb R^3\) defined by \(T(x_1,x_2,x_3)=(x_1,x_2,5)\). Since \(T(0,0,0)=(0,0,5) \neq (0,0,0)\), \(T\) is not a linear transformation.

Theorem. For a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), there exists a unique \(m\times n\) matrix \(A\), called the standard matrix of \(T\), for which \[T(\overrightarrow{x})=A\overrightarrow{x} \text{ for all } \overrightarrow{x}\in \mathbb R^n.\] Moreover, \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \:\cdots T(\overrightarrow{e_n})]\) where \(\overrightarrow{e_i}\) is the \(i\)th column of \(I_n\).

Example.

1. Use the standard matrix to find the rotation transformation \(T:\mathbb R^2 \to \mathbb R^2\) that rotates each point of \(\mathbb R^2\) about the origin through an angle \(\theta\) counterclockwise.
   
   Solution. By trigonometry we have \[T(\overrightarrow{e_1})=T\left(\left[\begin{array}{c}1\\0 \end{array} \right] \right)=\left[\begin{array}{c}\cos \theta \\ \sin\theta \end{array} \right] \text{ and } T(\overrightarrow{e_2})=T\left(\left[\begin{array}{c}0\\1 \end{array} \right] \right)=\left[\begin{array}{r} -\sin\theta\\ \cos \theta \end{array} \right].\] Then the standard matrix is \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} \cos \theta& -\sin\theta\\ \sin\theta& \cos \theta \end{array} \right].\) Thus \[T(\overrightarrow{x})=A\overrightarrow{x}, \text{ i.e., } T\left( \left[\begin{array}{c}x_1\\x_2 \end{array} \right] \right) =\left[\begin{array}{c}x_1\cos \theta -x_2\sin\theta \\ x_1\sin\theta +x_2\cos \theta \end{array} \right] \text{ for all } \overrightarrow{x}\in \mathbb R^2.\]


2. Consider the linear transformation \(T:\mathbb R^2 \to \mathbb R^3\) defined by \[T(x_1,x_2)=(x_1-x_2,2x_1+3x_2,4x_2).\] Note that \(T(\overrightarrow{e_1})=T(1,0)=(1,2,0)\) and \(T(\overrightarrow{e_2})=T(0,1)=(-1,3,4)\). The standard matrix of \(T\) is \[A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&-1\\ 2&3\\ 0&4 \end{array} \right].\]

For any given linear transformation \(T:\mathbb R^n \to \mathbb R^m\), the domain space is \(\mathbb R^n\) and the codomain space is \(\mathbb R^m\). We study a subspace of the domain space called Kernel or Null Space and a subspace of the codomain space called Image Space or Range.

Definition. The kernel or null space of a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), denoted by \(\ker (T)\) or \(\ker T\), is the following subspace of \(\mathbb R^n\): \[\ker T= \{\overrightarrow{x} \in \mathbb R^n \;|\; T(\overrightarrow{x})=\overrightarrow{0_m}\}.\] The nullity of \(T\), denoted by \(\operatorname{nullity}(T)\), is the dimension of \(\ker T\), i.e., \[\operatorname{nullity}(T)=\operatorname{dim}(\ker T).\]
Remark. If \(A\) is the standard matrix of a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), then \(\ker T=\operatorname{NS}(A)\) and \(\operatorname{nullity}(T)=\operatorname{nullity}(A)\).

Example. The linear transformation \(T:\mathbb R^3 \to \mathbb R^2\) defined by \(T(x_1,x_2,x_3)=(x_1,x_2)\) has the standard matrix \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \: T(\overrightarrow{e_3})] =\left[\begin{array}{rrr} 1&0&0\\ 0&1&0 \end{array} \right]\). Note that \[\ker T=\operatorname{NS}(A)=\operatorname{Span} \left\lbrace \left[\begin{array}{r} 0\\0\\1 \end{array} \right] \right\rbrace,\] and \(\operatorname{nullity}(T)=\operatorname{nullity}(A)=1\).

Definition. The image space or range of a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), denoted by \(\operatorname{im} (T)\) or \(\operatorname{im} T\) or \(T(\mathbb R^n)\), is the following subspace of \(\mathbb R^m\): \[\operatorname{im} T= \{T(\overrightarrow{x}) \;|\; \overrightarrow{x} \in \mathbb R^n\}.\] The rank of \(T\), denoted by \(\operatorname{rank}(T)\), is the dimension of \(\operatorname{im} T\), i.e., \[\operatorname{rank}(T)=\operatorname{dim}(\operatorname{im} T).\]

Remark. If \(A\) is the standard matrix of a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), then \(\operatorname{im} T=\operatorname{CS}\left(A\right)\) and \(\operatorname{rank}(T)=\operatorname{rank}(A)\).
Example. The linear transformation \(T:\mathbb R^2 \to \mathbb R^3\) defined by \(T(x_1,x_2)=(x_1,x_2,0)\) has the standard matrix \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&0\\ 0&1\\ 0&0 \end{array} \right]\). Note that \[\operatorname{im} T=\operatorname{CS}\left(A\right)=\operatorname{Span} \left\lbrace \left[\begin{array}{r} 1\\0\\0 \end{array} \right], \left[\begin{array}{r} 0\\1\\0 \end{array} \right] \right\rbrace,\] and \(\operatorname{rank}(T)=\operatorname{rank}(A)=2\).

Theorem.(Rank-Nullity Theorem) For a linear transformation \(T:\mathbb R^n \to \mathbb R^m\), \[\operatorname{rank}(T)+\operatorname{nullity}(T)=n.\]

Example. The linear transformation \(T:\mathbb R^3 \to \mathbb R^2\) defined by \(T(x_1,x_2,x_3)=(x_1,x_2)\) has \(\operatorname{nullity}(T)=1\) (see examples before). Then by the Rank-Nullity Theorem, \[\operatorname{rank}(T)=3-\operatorname{nullity}(T)=2.\]

Now we discuss two important types of linear transformation \(T:\mathbb R^n \to \mathbb R^m\).
Definition. Let \(T:\mathbb R^n \to \mathbb R^m\) be a linear transformation. \(T\) is onto if each \(\overrightarrow{b}\in \mathbb R^m\) has a pre-image \(\overrightarrow{x}\) in \(\mathbb R^n\) under \(T\), i.e., \(T(\overrightarrow{x})=\overrightarrow{b}\). \(T\) is one-to-one if each \(\overrightarrow{b}\in \mathbb R^m\) has at most one pre-image in \(\mathbb R^n\) under \(T\).

Example.

1. The linear transformation \(T:\mathbb R^3 \to \mathbb R^2\) defined by \(T(x_1,x_2,x_3)=(x_1,x_2)\) is onto because each \((x_1,x_2)\in \mathbb R^2\) has a pre-image \((x_1,x_2,0)\in \mathbb R^3\) under \(T\). But \(T\) is not one-to-one because \(T(0,0,0)=T(0,0,1)=(0,0)\), i.e., \((0,0)\) has two distinct pre-images \((0,0,0)\) and \((0,0,1)\) under \(T\).


2. The linear transformation \(T:\mathbb R^2 \to \mathbb R^3\) defined by \(T(x_1,x_2)=(x_1,x_2,0)\) is one-to-one because \(T(x_1,x_2)=T(y_1,y_2) \implies (x_1,x_2,0)=(x_1,x_2,0) \implies (x_1,x_2)=(y_1,y_2)\). But \(T\) is not onto because \((0,0,1)\in \mathbb R^3\) has no pre-image \((x_1,x_2)\in \mathbb R^2\) under \(T\).


3. The linear transformation \(T:\mathbb R^2 \to \mathbb R^2\) defined by \(T(x_1,x_2)=(x_1+x_2,x_1-x_2)\) is one-to-one and onto (exercise).

Theorem. Let \(T:\mathbb R^n \to \mathbb R^m\) be a linear transformation with the standard matrix \(A\). Then the following are equivalent.

1. \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is one-to-one.


2. \(\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_n}\}\).


3. \(\operatorname{nullity}(T)=\operatorname{nullity}(A)=0\).


4. The columns of \(A\) are linearly independent.

Example. The linear transformation \(T:\mathbb R^2 \to \mathbb R^3\) defined by \(T(x_1,x_2)=(x_1,x_2,0)\) has the standard matrix \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) ] =\left[\begin{array}{rr} 1&0\\ 0&1\\ 0&0 \end{array} \right]\). Note that the columns of \(A\) are linearly independent, \(\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_2}\}\), and \(\operatorname{nullity}(T)=\operatorname{nullity}(A)=0\). Thus \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is one-to-one.

Theorem. Let \(T:\mathbb R^n \to \mathbb R^m\) be a linear transformation with the standard matrix \(A\). Then the following are equivalent.

1. \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is onto.


2. \(\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^m\).


3. \(\operatorname{rank}(T)=\operatorname{rank}(A)=m\).


4. Each row of \(A\) has a pivot position.

Example. The linear transformation \(T:\mathbb R^3 \to \mathbb R^2\) defined by \(T(x_1,x_2,x_3)=(x_1,x_2)\) has the standard matrix \(A=[T(\overrightarrow{e_1})\: T(\overrightarrow{e_2}) \: T(\overrightarrow{e_3})] =\left[\begin{array}{rrr} 1&0&0\\ 0&1&0 \end{array} \right]\). Note that each row of \(A\) has a pivot position, \(\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^2\), and \(\operatorname{rank}(T)=\operatorname{rank}(A)=2\). Thus \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is onto.

Definition. A linear transformation \(T:\mathbb R^n \to \mathbb R^n\) is an isomorphism if it is one-to-one and onto.

Example. The linear transformation \(T:\mathbb R^2 \to \mathbb R^2\) defined by \(T(x_1,x_2)=(x_1+x_2,x_1-x_2)\) is one-to-one and onto consequently an isomorphism. Showing \(T\) is one-to-one is enough to show \(T\) is an isomorphism by the following theorem.

Theorem. Let \(T:\mathbb R^n \to \mathbb R^n\) be a linear transformation with the \(n\times n\) standard matrix \(A\). Then the following are equivalent.

1. \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is an isomorphism.


2. \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is one-to-one.


3. \(\ker T=\operatorname{NS}(A)=\{\overrightarrow{0_n}\}\).


4. \(\operatorname{nullity}(T)=\operatorname{nullity}(A)=0\).


5. The columns of \(A\) are linearly independent.


6. \(T\) (i.e., \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)) is onto.


7. \(\operatorname{im} T=\operatorname{CS}\left(A\right)=\mathbb R^n\).


8. \(\operatorname{rank}(T)=\operatorname{rank}(A)=n\).


9. Each row and column of \(A\) has a pivot position.

Example. What can we say about \(\operatorname{CS}\left(A\right),\operatorname{NS}(A),\operatorname{rank}(A), \operatorname{nullity}(A)\), and pivot positions of a \(3\times 3\) matrix with three linearly independent columns? What about \(\overrightarrow{x} \mapsto A\overrightarrow{x}\)?

Solution. By the preceding theorem, \(\operatorname{CS}\left(A\right)=\mathbb R^3,\operatorname{NS}(A) =\{\overrightarrow{0_3}\},\operatorname{rank}(A)=3,\operatorname{nullity}(A)=0\), \(A\) has 3 pivot positions, and \(\overrightarrow{x} \mapsto A\overrightarrow{x}\) is a one-to-one linear transformation from \(\mathbb R^3\) onto \(\mathbb R^3\).

Last edited