Row and column vectors

In linear algebra, a column vector with Template:Tmath elements is an <math>m \times 1</math> matrix<ref name="Artin">Template:Cite book</ref> consisting of a single column of Template:Tmath entries, for example, <math display="block">\boldsymbol{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}.</math>

Similarly, a row vector is a <math>1 \times n</math> matrix for some Template:Tmath, consisting of a single row of Template:Tmath entries, <math display="block">\boldsymbol a = \begin{bmatrix} a_1 & a_2 & \dots & a_n \end{bmatrix}. </math> (Throughout this article, boldface is used for both row and column vectors.)

The transpose (indicated by Template:Math) of any row vector is a column vector, and the transpose of any column vector is a row vector: <math display="block">\begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}</math> and <math display="block">\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix}^{\rm T} = \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}.</math>

The set of all row vectors with Template:Mvar entries in a given field (such as the real numbers) forms an Template:Mvar-dimensional vector space; similarly, the set of all column vectors with Template:Mvar entries forms an Template:Mvar-dimensional vector space.

The space of row vectors with Template:Mvar entries can be regarded as the dual space of the space of column vectors with Template:Mvar entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.

Notation

To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

<math display="block">\boldsymbol{x} = \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T}</math>

or

<math display="block">\boldsymbol{x} = \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}^{\rm T}</math>

Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below).Template:Fact

	Row vector	Column vector
Standard matrix notation (array spaces, no commas, transpose signs)	<math> \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix} </math>	<math> \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{bmatrix} \text{ or } \begin{bmatrix} x_1 \; x_2 \; \dots \; x_m \end{bmatrix}^{\rm T} </math>
Alternative notation 1 (commas, transpose signs)	<math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix} </math>	<math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix}^{\rm T} </math>
Alternative notation 2 (commas and semicolons, no transpose signs)	<math> \begin{bmatrix} x_1, x_2, \dots, x_m \end{bmatrix} </math>	<math> \begin{bmatrix} x_1; x_2; \dots; x_m \end{bmatrix} </math>

Operations

Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix.

The dot product of two column vectors Template:Math, considered as elements of a coordinate space, is equal to the matrix product of the transpose of Template:Math with Template:Math,

<math display="block">\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^\intercal \mathbf{b} = \begin{bmatrix}

   a_1  & \cdots  & a_n

\end{bmatrix} \begin{bmatrix}

   b_1 \\ \vdots \\ b_n

\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n \,, </math>

By the symmetry of the dot product, the dot product of two column vectors Template:Math is also equal to the matrix product of the transpose of Template:Math with Template:Math,

<math display="block">\mathbf{b} \cdot \mathbf{a} = \mathbf{b}^\intercal \mathbf{a} = \begin{bmatrix}

   b_1  & \cdots  & b_n

\end{bmatrix}\begin{bmatrix}

   a_1 \\ \vdots \\ a_n

\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n\,. </math>

The matrix product of a column and a row vector gives the outer product of two vectors Template:Math, an example of the more general tensor product. The matrix product of the column vector representation of Template:Math and the row vector representation of Template:Math gives the components of their dyadic product,

<math display="block">\mathbf{a} \otimes \mathbf{b} = \mathbf{a} \mathbf{b}^\intercal = \begin{bmatrix}

   a_1 \\ a_2 \\ a_3

\end{bmatrix}\begin{bmatrix}

   b_1 & b_2 & b_3

\end{bmatrix} = \begin{bmatrix} a_1 b_1 & a_1 b_2 & a_1 b_3 \\ a_2 b_1 & a_2 b_2 & a_2 b_3 \\ a_3 b_1 & a_3 b_2 & a_3 b_3 \\ \end{bmatrix} \,, </math>

which is the transpose of the matrix product of the column vector representation of Template:Math and the row vector representation of Template:Math,

<math display="block">\mathbf{b} \otimes \mathbf{a} = \mathbf{b} \mathbf{a}^\intercal = \begin{bmatrix}

   b_1 \\ b_2 \\ b_3

\end{bmatrix}\begin{bmatrix}

   a_1 & a_2 & a_3

\end{bmatrix} = \begin{bmatrix} b_1 a_1 & b_1 a_2 & b_1 a_3 \\ b_2 a_1 & b_2 a_2 & b_2 a_3 \\ b_3 a_1 & b_3 a_2 & b_3 a_3 \\ \end{bmatrix} \,. </math>

Matrix transformations

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An Template:Math matrix Template:Mvar can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector Template:Math, the product Template:Math is another row vector Template:Math:

<math display="block">\mathbf{v} M = \mathbf{p} \,.</math>

Another Template:Math matrix Template:Mvar can act on Template:Math,

<math display="block"> \mathbf{p} Q = \mathbf{t} \,. </math>

Then one can write Template:Math, so the matrix product transformation Template:Mvar maps Template:Math directly to Template:Math. Continuing with row vectors, matrix transformations further reconfiguring Template:Mvar-space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under an Template:Math matrix action, the operation occurs to the left,

<math display="block"> \mathbf{p}^\mathrm{T} = M \mathbf{v}^\mathrm{T} \,,\quad \mathbf{t}^\mathrm{T} = Q \mathbf{p}^\mathrm{T},</math>

leading to the algebraic expression Template:Math for the composed output from Template:Math input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.

See also

• Covariance and contravariance of vectors
• Index notation
• Vector of ones
• Single-entry vector
• Standard unit vector
• Unit vector

References

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