Vector spacesベクトル空間 and bases基底きてい

date2026-07-14document_iddoc_77e3ed64a7a767a25cf258db5e87bd86descriptionベクトル空間と基底を、体上の公理、生成、一次独立、一意表示の関係から整理し、座標表示が成立する理由を説明する講義である。prerequisitesベクトルの線型結合 / 連立一次方程式の基本type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

A vector spaceベクトル空間 is not only a collection of arrows. Functions, polynomials, sequences, and matrices can also be treated as vector spaces if addition and scalar multiplication are defined naturally and satisfy the required rules.

The purpose of this abstraction is to ignore the outward form of the objects and keep only the structure of adding, multiplying by scalars, and forming linear combinations. From that structure alone we can define span生成せいせい, linear independence一次独立いちじどくりつ, basis基底きてい, and dimension次元じげん.

2Terms and definitions

A vector spaceベクトル空間 $V$ over a field体たい $K$ is a set equipped with addition $u + v$ and scalar multiplication $c v$ satisfying the usual associativity, commutativity, distributivity, existence of a zero vector, and existence of additive inverses.

For vectors $v_{1}, . . ., v_{n}$ , a vector of the form

c_{1} v_{1} + \dots + c_{n} v_{n}

is a linear combination線型結合せんけいけつごう. The set of all such linear combinations is the span張る空間.

The vectors $v_{1}, . . ., v_{n}$ are linearly independent一次独立いちじどくりつ when

c_{1} v_{1} + \dots + c_{n} v_{n} = 0

forces

c_{1} = \dots = c_{n} = 0.

A list of vectors that spans $V$ and is linearly independent is a basis基底きてい of $V$ .

3Plan

To understand a basis, separate the two ideas “enough” and “no redundancy.”

The vectors $v_{1}, . . ., v_{n}$ span $V$ when every vector in $V$ can be made from them. This is the “enough” condition. Linear independence means that none of the listed vectors can be made from the others. This is the “no redundancy” condition.

A basis基底きてい is a minimal coordinate system satisfying both conditions at once.

4Intuitive explanation

4.11. The plane as an example

In $R^{2}$ , using

e_{1} = (\begin{matrix} 1 \\ 0 \end{matrix}), e_{2} = (\begin{matrix} 0 \\ 1 \end{matrix}),

any vector

(\begin{matrix} x \\ y \end{matrix})

can be written as

(\begin{matrix} x \\ y \end{matrix}) = x e_{1} + y e_{2.}

Moreover, the coefficients $x, y$ are unique. This is the meaning of coordinates.

4.22. A basis determines a way to measure

The same vector can have different coordinates when the basis changes. The vector itself is an object in the space; coordinates are the numbers obtained by measuring that object using a basis. This distinction becomes important in change of basis基底変換きていへんかん and similarity相似そうじ.

5Formal explanation

5.11. Coordinate expression by a basis

When $B = (v_{1}, . . ., v_{n})$ is a basis of $V$ , every $v i n V$ has a unique expression

v = x_{1} v_{1} + \dots + x_{n} v_{n} .

The list of coefficients is called the coordinates座標ざひょう of $v$ with respect to $B$ .

5.22. Proof of uniqueness

Suppose

v = x_{1} v_{1} + \dots + x_{n} v_{n} = y_{1} v_{1} + \dots + y_{n} v_{n} .

Subtracting the two expressions gives

0 = (x_{1} - y_{1}) v_{1} + \dots + (x_{n} - y_{n}) v_{n} .

Because the basis is linearly independent,

x_{1} - y_{1} = \dots = x_{n} - y_{n} = 0.

Thus $x_{i} = y_{i}$ for every $i$ .

Only subtraction is used here. No coefficient is divided by, so we do not need to check whether a particular coefficient is nonzero.

6Worked example: deciding whether vectors form a basis

6.1Problem

Decide whether the two vectors in $R^{2}$

v_{1} = (\begin{matrix} 1 \\ 2 \end{matrix}), v_{2} = (\begin{matrix} 3 \\ 4 \end{matrix})

form a basis of $R^{2}$ .

6.2Explanation

For two vectors in $R^{2}$ , being a basis is equivalent to being linearly independent. Consider coefficients $a, b$ satisfying

a (\begin{matrix} 1 \\ 2 \end{matrix}) + b (\begin{matrix} 3 \\ 4 \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) .

Componentwise, this gives

a + 3 b = 0, 2 a + 4 b = 0.

From the first equation, $a = - 3 b$ . Substituting this into the second gives

2 (- 3 b) + 4 b = - 2 b = 0,

so $b = 0$ , and therefore $a = 0$ . Hence the vectors are linearly independent and form a basis of $R^{2}$ .

This example shows that basis checking is the task of confirming that there is no redundant vector. Later, when we inspect the columns of a matrix, we use the same idea to test linear independence of column vectors.

7What changes over real and complex scalars

The same underlying set can become a different vector space when the scalar field changes. For example, $C$ is one-dimensional over $C$ , but two-dimensional over $R$ .

This distinction matters when studying complex inner products複素内積ふくそないせき, the Hermitian propertyエルミート性, and unitary matricesユニタリ行列. If the scalar field is unclear, the meaning of homogeneity changes.

8Theorem and proof idea: finite bases have the same size

8.1Theorem

Every basis of a finite-dimensional vector space $V$ has the same number of vectors. This number is the dimension次元じげん of $V$ .

8.2Proof idea

A full proof uses the exchange lemma. The intuition is this: if $m$ linearly independent vectors lie in a space spanned by $n$ vectors, then $m < = n$ . One cannot create more independent directions than the number of generators available.

Let $B$ and $C$ be two bases, with $m$ and $n$ vectors respectively. Since $B$ is linearly independent and $C$ spans the space, $m < = n$ . Conversely, since $C$ is linearly independent and $B$ spans the space, $n < = m$ . Therefore $m = n$ .

9Final form

A basis基底きてい is a list of vectors that both generates the whole space and has no redundancy.

basis = spanningset + linearindependence .

Once a basis is chosen, vectors can be represented as lists of numbers. Thus a basis is also the coordinate system that lets us represent linear maps by matrices.

10Exercise links

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