Row Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Vectors #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

A row matrix (row vector) is a matrix with exactly one row and $n$ columns, i.e., a $1 \times n$ matrix.

◇Key Concepts

Dimensions: $1 \times n$ Represents a vector in $\mathbb{R}^n$ written horizontally Transpose of a row matrix is a ◇Column Matrix Used in dot product computations

◇Details / Theory

Definition

A row matrix has the form:

$\mathbf{v}^T = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}$

where $v_j \in \mathbb{R}$ for $j = 1, 2, \ldots, n$.

Example

$\mathbf{r} = \begin{bmatrix} 2 & -5 & 0 & 4 \end{bmatrix}$

This is a $1 \times 4$ row matrix.

Dot Product as Row × Column

The dot product of two vectors can be expressed as a row matrix times a column matrix:

$\mathbf{u} \cdot \mathbf{v} = \mathbf{u}^T \mathbf{v} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = u_1 v_1 + u_2 v_2 + u_3 v_3$

Relationship to Column Matrix

$\mathbf{r}^T$ converts a row matrix to a ◇Column Matrix and vice versa.

◇Common Pitfalls

Confusing $1 \times n$ (row) with $n \times 1$ (column) — order matters. Attempting to add a row matrix and column matrix of the same number of entries — dimensions don't match.

◇Summary

TL;DR — A row matrix is a $1 \times n$ matrix (one row, $n$ columns). Its transpose is a column matrix. Row-times-column multiplication yields the dot product.

◇Links

Related: ◇Column Matrix — ◇Definition of a Matrix — ◇Size & Dimension of a Matrix — ◇Transpose of a Matrix — ◇Types of Matrices Source / Reference:

◇References & Navigation

• Previous Topic: ◇Properties of Matrix Operations
• Next Topic: ◇Scalar Matrix
• Related Notes: ◇Important Remarks, ◇Scalar Multiplication, ◇Matrices Detailed