Column Matrix

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

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◇Overview

A column matrix (column vector) is a matrix with exactly one column and $m$ rows, i.e., an $m \times 1$ matrix.

◇Key Concepts

Dimensions: $m \times 1$ Represents a vector in $\mathbb{R}^m$ Used in matrix-vector multiplication $A\mathbf{x} = \mathbf{b}$ Transpose of a column matrix is a ◇Row Matrix

◇Details / Theory

Definition

A column matrix has the form:

$\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix}$

where $v_i \in \mathbb{R}$ for $i = 1, 2, \ldots, m$.

Notation

Often denoted with bold lowercase: $\mathbf{v}$, $\mathbf{x}$, $\mathbf{b}$ In systems of equations, the solution vector and constant vector are column matrices

Example

$\mathbf{x} = \begin{bmatrix} 3 \\ -1 \\ 7 \end{bmatrix}$

This is a $3 \times 1$ column matrix (a vector in $\mathbb{R}^3$).

Relationship to Row Matrix

If $\mathbf{v}$ is a column matrix, then $\mathbf{v}^T$ is a ◇Row Matrix:

$\begin{bmatrix} 3 \\ -1 \\ 7 \end{bmatrix}^T = \begin{bmatrix} 3 & -1 & 7 \end{bmatrix}$

◇Common Pitfalls

Confusing $m \times 1$ (column) with $1 \times n$ (row) — dimensions are rows × columns. Forgetting that a column vector and row vector with the same entries are NOT equal as matrices.

◇Summary

TL;DR — A column matrix is an $m \times 1$ matrix representing a vector. Its transpose is a row matrix. Column vectors are fundamental in linear systems $A\mathbf{x} = \mathbf{b}$.

◇Links

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

◇References & Navigation

• Next Topic: ◇Definition of a Matrix
• Related Notes: ◇Properties of Matrix Operations, ◇Trace of a Matrix, ◇Matrices Detailed