Essential Vocabulary

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Composition of linear transformations

Let \(U\), \(V\) and \(W\) be vector spaces, and let \(T:U\rightarrow V\) and \(S:V\rightarrow W\) be linear transformations. The composition of \(S\) and \(T\) is the transformation \(S\circ T:U\rightarrow W\) given by
\[(S\circ T)(\vec {u})=S(T(\vec {u}))\]

The matrix of a composition is the product of the matrices corresponding to the transformations in the composition, in the same order.

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Image of a linear transformation

Let \(V\) and \(W\) be vector spaces, and let \(T:V\rightarrow W\) be a linear transformation. The image of \(T\), denoted by \(\mbox {im}(T)\), is the set
\[\mbox {im}(T)=\{T(\vec {v}):\vec {v}\in V\}\]
In other words, the image of \(T\) consists of individual images of all vectors of \(V\).

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Inverse of a linear transformation

Let \(V\) and \(W\) be vector spaces, and let \(T:V\rightarrow W\) be a linear transformation. A transformation \(S:W\rightarrow V\) that satisfies \(S\circ T=\id _V\) and \(T\circ S=\id _W\) is called an inverse of \(T\). If \(T\) has an inverse, \(T\) is called invertible.

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Kernel of a linear transformation

Let \(V\) and \(W\) be vector spaces, and let \(T:V\rightarrow W\) be a linear transformation. The kernel of \(T\), denoted by \(\mbox {ker}(T)\), is the set
\[\mbox {ker}(T)=\{\vec {v}:T(\vec {v})=\vec {0}\}\]
In other words, the kernel of \(T\) consists of all vectors of \(V\) that map to \(\vec {0}\) in \(W\).

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Linear transformation

A transformation \(T:\RR ^n\rightarrow \RR ^m\) is called a linear transformation if the following are true for all vectors \(\vec {u}\) and \(\vec {v}\) in \(\RR ^n\), and scalars \(k\).
\[T(k\vec {u})= kT(\vec {u})\]
\[T(\vec {u}+\vec {v})= T(\vec {u})+T(\vec {v})\]

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Standard matrix of a linear transformation

Let \(T:\RR ^n\rightarrow \RR ^m\) be a linear transformation. Then the matrix
\[A=\begin{bmatrix} | & |& &|\\ T(\vec {e}_1) & T(\vec {e}_2)&\dots &T(\vec {e}_n)\\ |&| & &| \end{bmatrix} \]
is known as the standard matrix of the linear transformation \(T\).

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