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

$\mathbf {V} ={\begin{bmatrix}{\begin{array}{cccccc}v_{1}\\v_{2}\\\vdots \\v_{n}\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{cccccc}v_{1}\\v_{2}\\\vdots \\v_{n}\end{array}}\end{matrix}}\right),$

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

$\mathbf {W} =\mathbf {V} ^{\rm {T}}={\begin{bmatrix}{\begin{array}{cccccc}v_{1}&v_{2}&\cdots &v_{n}\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{cccccc}v_{1}&v_{2}&\cdots &v_{n}\end{array}}\end{matrix}}\right),$

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

An example of a column vector with 5 entries is

${\begin{bmatrix}{\begin{array}{rrrrr}9\\-5\\7\\1\\19\end{array}}\end{bmatrix}},$

while an example of a row vector with 5 entries is

${\begin{bmatrix}{\begin{array}{rrrrr}9&-5&7&1&19\end{array}}\end{bmatrix}}.$

See also

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