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The action of \(T\) on all of the elements of \(\RR ^n\) is completely determined by where \(T\) maps the standard unit vectors. (See
Examples ex:imageOfBasisVectors and ex:imageofatransformation)
The last point in the summary is so important that it is worth illustrating again.
Let \(T:\RR ^3\rightarrow \RR ^2\) be a linear transformation. Suppose that the only information we have about this transformation is that \(T(\vec {i})=\begin{bmatrix}3\\-1\end{bmatrix}\), \(T(\vec {j})=\begin{bmatrix}0\\4\end{bmatrix}\) and \(T(\vec {k})=\begin{bmatrix}-2\\1\end{bmatrix}\). Is this
information sufficient to determine the image of \(\vec {w}=\begin{bmatrix}1\\-3\\6\end{bmatrix}\)?
Observe that
\[\vec {w}=\vec {i}-3\vec {j}+6\vec {k}\]
We find \(T(\vec {w})\) by using the fact that \(T\) is linear.
Because of properties of linear transformations, the information about the images of the standard unit vectors proved to be
sufficient for us to determine the image of \(\vec {w}\).
In Example ex:imageofatransformation, there was nothing special about the vector \(\vec {w}\). Any vector \(\vec {x}\) of \(\RR ^n\) can be written as a unique linear combination of the
standard unit vectors \(\vec {e}_1,\ldots , \vec {e}_n\). Therefore, the image of any vector \(\vec {x}\) under a linear transformation \(T:\RR ^n\rightarrow \RR ^m\) is uniquely determined by the
images of \(\vec {e}_1, \ldots , \vec {e}_n\). Knowing \(T(\vec {e}_1),\ldots , T(\vec {e}_n)\) allows us to construct a matrix \(A\), with \(T(\vec {e}_1),\ldots , T(\vec {e}_n)\) as columns, that induces transformation \(T\). We formalize this idea in
a theorem.
Let \(T:\RR ^n\rightarrow \RR ^m\) be a linear transformation. Then \(T\) is a matrix transformation with
Thus, for every \(\vec {x}\) in \(\RR ^n\), we have \(T(\vec {x})=A\vec {x}\).
Theorem th:matrixtran shows that every matrix transformation is linear. Theorem th:matlin states that every linear transformation from \(\RR ^n\) into \(\RR ^m\) is a
matrix transformation. We combine these results in a corollary.
A transformation \(T:\RR ^n\rightarrow \RR ^m\) is a linear transformation if and only if it is a matrix transformation.
The results of this section rely on the fact that every vector of \(\RR ^n\) can be written as a unique linear combination
of the standard unit vectors \(\vec {e}_1,\vec {e}_2,\dots ,\vec {e}_n\). These vectors form the standard basis for \(\RR ^n\). We will see in Matrix of a Linear
Transformation with Respect to Arbitrary Bases that the matrix used to represent a linear transformation depends
on a choice of basis. Because we are using the standard basis, it is natural to name the matrix in Theorem th:matlin
accordingly.
The matrix in Theorem th:matlin is known as the standard matrix of the linear transformation\(T\).
The standard matrix of a linear transformation \(T:\RR ^3\rightarrow \RR ^2\) such that \(T(\vec {i})=\begin{bmatrix}2\\-1\end{bmatrix}\), \(T(\vec {j})=\begin{bmatrix}-1\\3\end{bmatrix}\) and \(T(\vec {k})=\begin{bmatrix}0\\4\end{bmatrix}\) is
\[A=\begin{bmatrix}2&-1&0\\-1&3&4\end{bmatrix}\]
Find the standard matrix of a linear transformation \(T:\RR ^2\rightarrow \RR ^2\) such that \(T(\vec {i})=2\vec {i}\) and \(T(\vec {j})=2\vec {j}\).
We use the images of \(\vec {i}\) and \(\vec {j}\) as columns of the matrix. The
standard matrix of \(T\) is
\[\begin{bmatrix}2&0\\0&2\end{bmatrix}\]
Find the standard matrix of a linear transformation \(T:\RR ^2\rightarrow \RR ^4\) if \(T\left (\begin{bmatrix}3\\1\end{bmatrix}\right )=\begin{bmatrix}6\\1\\13\\-1\end{bmatrix}\) and \(T\left (\begin{bmatrix}-2\\0\end{bmatrix}\right )=\begin{bmatrix}-2\\0\\-8\\2\end{bmatrix}\).
In this example we are not given the images of the standard basis
vectors \(\vec {i}\) and \(\vec {j}\). However, we can find the images of \(\vec {i}\) and \(\vec {j}\) by expressing \(\vec {i}\) and \(\vec {j}\) as linear combinations of \(\begin{bmatrix}3\\1\end{bmatrix}\) and \(\begin{bmatrix}-2\\0\end{bmatrix}\), then apply the
fact that \(T\) is linear.
Suppose that a linear transformation \(T:\RR ^2\rightarrow \RR ^3\) is such that \(T(\vec {i})=\begin{bmatrix}-4\\2\\1\end{bmatrix}\) and \(T(\vec {j})=\begin{bmatrix}0\\-1\\5\end{bmatrix}\). Find \(T\Big (\begin{bmatrix}4\\-1\end{bmatrix}\Big )\).
Suppose that a linear transformation \(T:\RR ^2\rightarrow \RR ^3\) is such that \(T\Big (\begin{bmatrix}1\\-1\end{bmatrix}\Big )=\begin{bmatrix}1\\4\\-1\end{bmatrix}\) and \(T\Big (\begin{bmatrix}2\\0\end{bmatrix}\Big )=\begin{bmatrix}0\\6\\4\end{bmatrix}\). Find the standard matrix \(A\) of \(T\).
Find the standard matrix \(A\) of the linear transformation \(T:\RR ^2\rightarrow \RR ^2\) if \(T\) doubles the \(x\) component of every vector and triples the \(y\)
component.
Find the standard matrix \(A\) of the linear transformation \(T:\RR ^2\rightarrow \RR ^2\) if \(T\) projects each vector onto the \(x\)-axis. (e.g. \(T\left (\begin{bmatrix}4\\5\end{bmatrix}\right )=\begin{bmatrix}4\\0\end{bmatrix}\))
Find the standard matrix \(A\) of the linear transformation \(T:\RR ^2\rightarrow \RR ^2\) if \(T\) projects each vector onto the \(y\)-axis. (e.g. \(T\left (\begin{bmatrix}4\\5\end{bmatrix}\right )=\begin{bmatrix}0\\5\end{bmatrix}\))
Week 8