Essential Vocabulary

Here is a link to a list of these terms on Quizlet

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Basis

A set \(\mathcal {S}\) of vectors is called a basis of \(\RR ^n\) (or a basis of a subspace \(V\) of \(\RR ^n\)) provided that
1.
\(\mbox {span}(\mathcal {S})=\RR ^n\) (or \(V\))
2.
\(\mathcal {S}\) is linearly independent.

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Closed under addition

A set \(V\) is said to be closed under addition if for each element \(\vec {u} \in V\) and \(\vec {v} \in V\) the sum \(\vec {u}+\vec {v}\) is also in \(V\).

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Closed under scalar multiplication

A set \(V\) is said to be closed under scalar multiplication if for each element \(\vec {v} \in V\) and for each scalar \(k \in \RR \) the product \(k\vec {v}\) is also in \(V\).

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Column space of a matrix

Let \(A\) be an \(m\times n\) matrix. The column space of \(A\), denoted by \(\mbox {col}(A)\), is the subspace of \(\RR ^m\) spanned by the columns of \(A\).

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Coordinate vector with respect to a basis

Let \(\mathcal {B} = \{\vec {v}_1, \vec {v}_2,\ldots ,\vec {v}_k\}\) be an ordered basis. Then the coordinate vector \(\vec {v}\) is the column vector \(\begin{bmatrix}c_1\\ c_2\\ \vdots \\c_k\end{bmatrix}\) such that \(\vec {v} = c_1\vec {v}_1+c_2\vec {v}_2+\ldots +c_p\vec {v}_k\).

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Dimension

Let \(V\) be a subspace of \(\RR ^n\). The dimension of \(V\) is the number, \(m\), of elements in any basis of \(V\). We write
\[\mbox {dim}(V)=m\]

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Null space of a matrix

Let \(A\) be an \(m\times n\) matrix. The null space of \(A\), denoted by \(\mbox {null}(A)\), is the set of all vectors \(\vec {x}\) in \(\RR ^n\) such that \(A\vec {x}=\vec {0}\). It is a subspace of \(\RR ^n\).

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Nullity of a matrix

Let \(A\) be a matrix. The dimension of the null space of \(A\) is called the nullity of \(A\).
\[\mbox {dim}\Big (\mbox {null}(A)\Big )=\mbox {nullity}(A)\]

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Ordered basis

A basis in which the elements appear in a specific fixed order. Establishing an order is necessary because a coordinate vector with respect to a given basis relies on the order in which the basis elements appear.

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Rank of a matrix

Let \(A\) be a matrix. The dimension of the row space of \(A\) is called the rank of \(A\).
\[\mbox {dim}\Big (\mbox {row}(A)\Big )=\mbox {rank}(A)\]

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Rank-Nullity theorem

Let \(A\) be an \(m\times n\) matrix. Then
\[\mbox {rank}(A)+\mbox {nullity}(A)=n\]

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Row space of a matrix

Let \(A\) be an \(m\times n\) matrix. The row space of \(A\), denoted by \(\mbox {row}(A)\), is the subspace of \(\RR ^n\) spanned by the rows of \(A\).

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Subspace

Suppose that \(V\) is a nonempty subset of \(\RR ^n\) that is closed under addition and closed under scalar multiplication. Then \(V\) is a subspace of \(\RR ^n\).

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