Basic operations on vectorsベクトル

date2026-07-14document_iddoc_3365b297d0d255378781b8c8569bbcf8descriptionベクトルを成分をもつ計算対象として導入し、和・差・スカラー倍・線型結合から線型代数の基礎操作へ接続する講義である。prerequisites数の四則演算 / 座標平面の基本type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

The essential point is to treat a vectorベクトル not only as an arrow but also as an object with ordered components成分せいぶん that can be calculated. Vector addition, subtraction, scalar multiplicationスカラー倍, and linear combinations線型結合せんけいけつごう are the entry point to linear algebra.

data/lecture/math/vector/introduction-to-vectors.lecture.n.md

2Terms and definitions

A vector in $R^{n}$ is an ordered list of numbers, written as

v = (\begin{matrix} v_{1} \\ v_{2} \\ ⋮ \\ v_{n} \end{matrix}) .

The zero vector零ベクトル has all components equal to $0$ and is the additive identity. The standard basis標準基底ひょうじゅんきてい of $R^{n}$ consists of vectors with one component equal to $1$ and the others equal to $0$ .

A column vector列れつベクトル lists components vertically, while a row vector行ぎょうベクトル lists them horizontally. Shape matters when multiplying by matrices.

A scalarスカラー is a coefficient used to scale a vector. A linear combination線型結合せんけいけつごう has the form

c_{1} v_{1} + \dots + c_{k} v_{k} .

3Strategy

First check component-wise calculations in two dimensions次元じげん, then extend the same rule to $n$ dimensions. In linear algebra, increasing the dimension does not change the rule: each coordinate is handled in the same way.

4Intuitive explanation

Adding vectors corresponds to performing displacements one after another. Scalar multiplicationスカラー倍 changes the amount of displacement in the same direction.

In component notation, these operations are carried out separately on each coordinate. Thus vector operations are not a new complicated rule; they are parallel calculations along the coordinate axes.

5Precise explanation

For $u = (u_{1}, \dots, u_{n})^{T}$ and $v = (v_{1}, \dots, v_{n})^{T}$ , vector addition and subtraction are defined by

u + v = (u_{1} + v_{1}, \dots, u_{n} + v_{n})^{T},

u - v = (u_{1} - v_{1}, \dots, u_{n} - v_{n})^{T} .

For a scalar $c$ ,

c u = (c u_{1}, \dots, c u_{n})^{T} .

These definitions make addition and scalar multiplication work together. For example,

c_{1} v_{1} + c_{2} v_{2}

is a linear combination線型結合せんけいけつごう of two vectors with chosen coefficients係数けいすう.

6Worked example

Let $u = (1, 2)^{T}$ and $v = (3, - 1)^{T}$ . Then

u + v = (4, 1)^{T}, u - v = (- 2, 3)^{T},

and

2 u - 3 v = 2 (1, 2)^{T} - 3 (3, - 1)^{T} = (- 7, 7)^{T} .

The first and second components are processed independently. Also, if $e_{1} = (1, 0)^{T}$ and $e_{2} = (0, 1)^{T}$ , then $(a, b)^{T} = a e_{1} + b e_{2}$ .

7Common misunderstandings

Do not identify a vector with a point without context. A point is a position; a vector represents displacement or components.
Increasing the dimension does not change the rules: addition and scalar multiplication remain component-wise.
A linear combination線型結合せんけいけつごう is not just ordinary addition; the coefficients are freely chosen and determine how directions are combined.

8Scope

This page uses real scalars. With complex scalars the basic operations have the same form, but inner products and angles require conjugation.

9Final forms

[PARSE ERROR: Undefined("Command(\"boxed\")")] u + v = (u_{1} + v_{1}, \dots, u_{n} + v_{n})^{T}

[PARSE ERROR: Undefined("Command(\"boxed\")")] c u = (c u_{1}, \dots, c u_{n})^{T}

[PARSE ERROR: Undefined("Command(\"boxed\")")] c_{1} v_{1} + \dots + c_{k} v_{k} isalinearcombination

[PARSE ERROR: Undefined("Command(\"boxed\")")] v = v_{1} e_{1} + \dots + v_{n} e_{n} (standardbasis)

10In one sentence

Basic vector operations are performed component by component, and linear combinations線型結合せんけいけつごう become the entry point to bases and rank階数かいすう.

11Exercise links

data/exercise/math/linear-algebra/vectors-and-linear-combinations.exercise.n.md