Essential Vocabulary

Coordinate vector

Let \(V\) be a vector space, and let \(\mathcal {B}=\{\vec {v}_1, \ldots ,\vec {v}_n\}\) be a basis for \(V\). If \(\vec {v}=a_1\vec {v}_1+\ldots +a_n\vec {v}_n\), then the vector in \(\RR ^n\) whose components are the coefficients \(a_1, \ldots ,a_n\) is said to be the coordinate vector for \(\vec {v}\) with respect to \(\mathcal {B}\). We denote the coordinate vector by \([\vec {v}]_{\mathcal {B}}\) and write:
\[[\vec {v}]_{\mathcal {B}}=\begin{bmatrix}a_1\\\vdots \\a_n\end{bmatrix}\]

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Finite-dimensional vector space

A vector space is said to be finite-dimensional if it is spanned by finitely many vectors.

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Isomorphism

Let \(V\) and \(W\) be vector spaces. If there exists an invertible linear transformation \(T:V\rightarrow W\) we say that \(V\) and \(W\) are isomorphic and write \(V\cong W\). The invertible linear transformation \(T\) is called an isomorphism.

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Matrix of a linear transformation

Let \(V\) and \(W\) be finite-dimensional vector spaces with ordered bases \(\mathcal {B}=\{\vec {v}_1,\vec {v}_2,\ldots ,\vec {v}_n\}\) and \(\mathcal {C}\), respectively. Suppose \(T:V\rightarrow W\) is a linear transformation.
\[\text {Let}\quad A=\begin{bmatrix} | & |& &|\\ [T(\vec {v}_1)]_{\mathcal {C}} & [T(\vec {v}_2)]_{\mathcal {C}}&\dots &[T(\vec {v}_n)]_{\mathcal {C}}\\ |&| & &| \end{bmatrix}\]
Then \(A[\vec {v}]_{\mathcal {B}}=[T(\vec {v})]_{\mathcal {C}}\) for all vectors \(\vec {v}\) in \(V\).

Matrix \(A\) is called the matrix of \(T\) with respect to ordered bases \(\mathcal {B}\) and \(\mathcal {C}\).

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One-to-one linear transformation

A linear transformation \(T:V\rightarrow W\) is one-to-one if
\[T(\vec {v}_1)=T(\vec {v}_2)\quad \text {implies that}\quad \vec {v}_1=\vec {v}_2\]

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Onto linear transformation

A linear transformation \(T:V\rightarrow W\) is onto if for every element \(\vec {w}\) of \(W\), there exists an element \(\vec {v}\) of \(V\) such that \(T(\vec {v})=\vec {w}\).

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Subspace

A nonempty subset \(U\) of a vector space \(V\) is called a subspace of \(V\), provided that \(U\) is itself a vector space when given the same addition and scalar multiplication as \(V\).

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Vector Space

Let \(V\) be a nonempty set. Suppose that elements of \(V\) can be added together and multiplied by scalars. The set \(V\), together with operations of addition and scalar multiplication, is called a vector space provided that
  • \(V\) is closed under addition
  • \(V\) is closed under scalar multiplication

and the following properties hold for \(\vec {u}\), \(\vec {v}\) and \(\vec {w}\) in \(V\) and scalars \(k\) and \(p\):

1.
Commutative Property of Addition: \(\vec {u}+\vec {v}=\vec {v}+\vec {u}\)
2.
Associative Property of Addition: \((\vec {u}+\vec {v})+\vec {w}=\vec {u}+(\vec {v}+\vec {w})\)
3.
Existence of Additive Identity: \(\vec {u}+\vec {0}=\vec {u}\)
4.
Existence of Additive Inverse: \(\vec {u}+(-\vec {u})=\vec {0}\)
5.
Distributive Property over Vector Addition: \(k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}\)
6.
Distributive Property over Scalar Addition: \((k+p)\vec {u}=k\vec {u}+p\vec {u}\)
7.
Associative Property for Scalar Multiplication: \(k(p\vec {u})=(kp)\vec {u}\)
8.
Multiplication by \(1\): \(1\vec {u}=\vec {u}\)

We will refer to elements of \(V\) as vectors.

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Note that definitions of span, linear independence, basis, and dimension are analogous to those for subspaces of \(\RR ^n\).