Row vector in the context of Square matrix

⭐ Core Definition: Row vector

In linear algebra, a column vector with ⁠ $m$ ⁠ elements is an $m\times 1$ matrix consisting of a single column of ⁠ $m$ ⁠ entries. Similarly, a row vector is a $1\times n$ matrix, consisting of a single row of ⁠ $n$ ⁠ entries. For example, ⁠ ${\boldsymbol {x}}$ ⁠ is a column vector and ⁠ ${\boldsymbol {a}}$ ⁠ is a row matrix:

${\boldsymbol {x}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}},\quad {\boldsymbol {a}}={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\end{bmatrix}}.$ (Throughout this article, boldface is used for both row and column vectors.)

↓ Menu

HINT:

👉 Row vector in the context of Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order $n$ . Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if $R$ is a square matrix representing a rotation (rotation matrix) and $\mathbf {v}$ is a column vector describing the position of a point in space, the product $R\mathbf {v}$ yields another column vector describing the position of that point after that rotation. If $\mathbf {v}$ is a row vector, the same transformation can be obtained using $\mathbf {v} R^{\mathsf {T}}$ , where $R^{\mathsf {T}}$ is the transpose of $R$ .

↓ Explore More Topics

In this Dossier

⭐ Core Definition: Row vector
👉 Row vector in the context of Square matrix
Row vector in the context of Transformation matrix
Row vector in the context of Coordinate vector

Row vector in the context of Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If $T$ is a linear transformation mapping $\mathbb {R} ^{n}$ to $\mathbb {R} ^{m}$ and $\mathbf {x}$ is a column vector with $n$ entries, then there exists an $m\times n$ matrix $A$ , called the transformation matrix of $T$ , such that: $T(\mathbf {x} )=A\mathbf {x}$ Note that $A$ has $m$ rows and $n$ columns, whereas the transformation $T$ is from $\mathbb {R} ^{n}$ to $\mathbb {R} ^{m}$ . There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

View the full Wikipedia page for Transformation matrix

↑ Return to Menu

Row vector in the context of Coordinate vector

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations.

The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below.

View the full Wikipedia page for Coordinate vector

↑ Return to Menu

Row vector in the context of Square matrix

Row vector Study page number 1 of 1

Play TriviaQuestions Online!

Skip to study material about Row vector in the context of "Square matrix"

⭐ Core Definition: Row vector

👉 Row vector in the context of Square matrix

Row vector in the context of Transformation matrix

Row vector in the context of Coordinate vector