Vectors

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A vector is a linear array of numbers, symbols, or expressions, arranged either in rows (row vectors) or columns (column vectors). (A vector is a tensor of rank 1.) The individual items in a vector are called its elements or entries.

An $n$ vector $\mathbf{𝐕}$ is commonly written using box brackets (an alternative notation uses large parentheses instead). An $n$ column vector $\mathbf{𝐕}$ is written

$\mathbf{𝐕}=\left[\begin{array}{c}\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\end{array}\right]=\left(\begin{array}{c}\begin{array}{c}{v}_1\\ v_2\\ ⋮\\ v_n\end{array}\end{array}\right),$

while an $n$ row vector $\mathbf{𝐖}$ is written

$\mathbf{𝐖}={\mathbf{𝐕}}^{\mathrm{T}}=\left[\begin{array}{c}\begin{array}{cccc}{v}_1& v_2& \cdots & v_n\end{array}\end{array}\right]=\left(\begin{array}{c}\begin{array}{cccc}{v}_1& v_2& \cdots & v_n\end{array}\end{array}\right),$

where ${\mathbf{𝐕}}^{\mathrm{T}}$ is the transpose of $\mathbf{𝐕}$.

An example of a column vector with 5 entries is

$\left[\begin{array}{c}\begin{array}{r}9\\ -5\\ 7\\ 1\\ 19\end{array}\end{array}\right],$

while an example of a row vector with 5 entries is

$\left[\begin{array}{c}\begin{array}{rrrrr}9& -5& 7& 1& 19\end{array}\end{array}\right].$

See also

• Scalars (tensors of rank 0)
• Vectors (tensors of rank 1)
• Matrices (tensors of rank 2)
• Tensors