Basics of linearity線型性せんけいせい

date2026-07-14document_iddoc_409a83bf1849062a03c7b6c414f30756description線型性を加法性と同次性から定義し、線型結合、基底ベクトルの像、行列の列へ接続する導入講義である。prerequisites高校ベクトルの和とスカラー倍 / 関数・写像の基本type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecture

1Introduction

Linearity線型性せんけいせい is the entrance to linear algebra線型代数せんけいだいすう. A map is linear when it is compatible with vectorベクトル addition and scalar multiplicationスカラー倍: doing the operation before the map gives the same result as mapping first and then doing the corresponding operation.

This viewpoint is placed first because later objects such as matrices行列ぎょうれつ, bases基底きてい, rank階数かいすう, the kernel核かく, and the image像ぞう are all understood through the same question: what does a linear map線型写像せんけいしゃぞう preserve, and what does it collapse?

2Terms and definition

Let $V$ and $W$ be vector spaces[ベクトル空間] over the same field体たい $K$ . A map $T : V - > W$ is a linear map線型写像せんけいしゃぞう when the following two conditions hold for every $u, v i n V$ and every $c i n K$ .

T (u + v) = T (u) + T (v)

T (c v) = c T (v)

The first condition is additivity加法性かほうせい; the second is homogeneity同次性どうじせい. Together they are called linearity線型性せんけいせい.

3Plan

In this lecture, linearity is not treated as a formula to memorize. We use the following order.

Additivity preserves how vectors are added, so the parallelogram rule is preserved.
Homogeneity preserves scalar stretching from the origin.
Therefore every linear combination線型結合せんけいけつごう is carried to the corresponding linear combination of images.
As a consequence, the whole map is determined by the images像ぞう of the basis基底きてい vectors.

With this order, the columns of a matrix naturally become the destinations of the basis vectors.

4Intuitive explanation

4.11. Linearity preserves the skeleton

A linear map線型写像せんけいしゃぞう preserves the straight structure measured from the origin. Operations such as bending, translating, or squaring are usually not linear, because they change the relationships created by addition and scalar multiplication.

4.22. Linear combinations are transported unchanged

The central consequence of linearity is that, for finitely many vectorsベクトル $v_{1}, . . ., v_{n}$ and scalars $c_{1}, . . ., c_{n}$ , we have

T (c_{1} v_{1} + \dots + c_{n} v_{n}) = c_{1} T (v_{1}) + \dots + c_{n} T (v_{n}) .

In words, mixing the ingredients first and then applying the map gives the same result as mapping the ingredients first and then mixing them with the same coefficients.

4.33. The images of basis vectors determine the whole map

If $e_{1}, . . ., e_{n}$ is a basis基底きてい, every vector $v$ has a unique expression

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

By linearity,

T (v) = x_{1} T (e_{1}) + \dots + x_{n} T (e_{n}) .

Therefore, once $T (e_{1}), . . ., T (e_{n})$ are known, $T (v)$ is known for every $v$ .

5Formal explanation

5.11. The zero vector is sent to the zero vector

For a linear map $T$ , the identity $T (0) = 0$ always holds. Substituting $c = 0$ into homogeneity gives, for any $v$ ,

T (0 v) = 0 T (v) .

The left side is $T (0)$ , and the right side is $0$ . No division is used here, so no extra nonzero condition is needed.

5.22. Preservation of linear combinations

For two terms this is exactly the definition. For several terms, repeat additivity and then use homogeneity.

T (c_{1} v_{1} + \dots + c_{n} v_{n}) = T (c_{1} v_{1}) + \dots + T (c_{n} v_{n}) = c_{1} T (v_{1}) + \dots + c_{n} T (v_{n}) .

This fact will be used later to explain the span張る空間, the column space列空間れつくうかん, and rank階数かいすう.

5.33. Determination by images of a basis

Let $e_{1}, . . ., e_{n}$ be a basis of $V$ . If two linear maps $S, T : V - > W$ agree on every basis vector, then for every $v = x_{1} e_{1} + \dots + x_{n} e_{n}$ ,

S (v) = x_{1} S (e_{1}) + \dots + x_{n} S (e_{n}) = x_{1} T (e_{1}) + \dots + x_{n} T (e_{n}) = T (v) .

Hence $S = T$ .

6Worked example: building a matrix from basis images

6.1Problem

Let $T : R^{2} - > R^{2}$ be a linear map satisfying

T (e_{1}) = (\begin{matrix} 2 \\ 1 \end{matrix}), T (e_{2}) = (\begin{matrix} - 1 \\ 3 \end{matrix}) .

Find the matrix of $T$ in the standard basis, and then find the image of $v = (4, 5)^{T}$ .

6.2Explanation

In the standard basis, the first column of the matrix is $T (e_{1})$ , and the second column is $T (e_{2})$ . Therefore

A = (\begin{matrix} 2 & - 1 \\ 1 & 3 \end{matrix}) .

Since $v = 4 e_{1} + 5 e_{2}$ , linearity gives

T (v) = 4 T (e_{1}) + 5 T (e_{2}) = 4 (\begin{matrix} 2 \\ 1 \end{matrix}) + 5 (\begin{matrix} - 1 \\ 3 \end{matrix}) = (\begin{matrix} 3 \\ 19 \end{matrix}) .

The same result is obtained by matrix multiplication.

A v = (\begin{matrix} 2 & - 1 \\ 1 & 3 \end{matrix}) (\begin{matrix} 4 \\ 5 \end{matrix}) = (\begin{matrix} 3 \\ 19 \end{matrix}) .

The point of this example is to view a matrix not merely as a table of numbers, but as a list of the images of the basis vectors. This viewpoint prepares for linear maps and matrices線型写像と行列, change of basis基底変換きていへんかん, and eigenvalues固有値こゆうち.

7What changes and what is preserved

A linear map preserves linear combinations, subspace structure, and the zero vector. It does not necessarily preserve length, angle, area, or dimension. For example, a projection can collapse information from a plane onto a line; it is linear, but it reduces dimension.

This distinction matters. An orthogonal matrix直交行列ちょっこうぎょうれつ or a unitary matrixユニタリ行列 is a special linear map that also preserves lengths and angles, while rank階数かいすう measures the number of directions that survive without being collapsed.

Projection, orthogonal matrices, and unitary matrices are defined later. Here they are only a preview of the fact that being linear does not determine which additional quantities an operation preserves.

8Theorem and proof: the kernel and image preserve linear structure

8.1Theorem

For a linear map $T : V - > W$ , $k e r T$ is a subspace of $V$ , and $i m T$ is a subspace of $W$ .

8.2Proof

First consider $k e r T = v i n V | T (v) = 0$ . Since $T (0) = 0$ , we have $0 i n k e r T$ . If $u, v i n k e r T$ , then

T (u + v) = T (u) + T (v) = 0 + 0 = 0,

so $u + v i n k e r T$ . Also, for $c i n K$ ,

T (c u) = c T (u) = c 0 = 0,

so $c u i n k e r T$ . Hence $k e r T$ is a subspace.

Next consider $i m T = T (v) | v i n V$ . Since $T (0) = 0$ , we have $0 i n i m T$ . If $y_{1} = T (v_{1})$ and $y_{2} = T (v_{2})$ , then

y_{1} + y_{2} = T (v_{1}) + T (v_{2}) = T (v_{1} + v_{2}),

so $y_{1} + y_{2} i n i m T$ . Similarly,

c y_{1} = c T (v_{1}) = T (c v_{1}),

so $c y_{1} i n i m T$ . Hence $i m T$ is a subspace.

9Final form

Linearity means satisfying additivity and homogeneity at the same time.

T (u + v) = T (u) + T (v), T (c v) = c T (v) .

The essence is that linear combinations are transported unchanged.

T (c_{1} v_{1} + \dots + c_{n} v_{n}) = c_{1} T (v_{1}) + \dots + c_{n} T (v_{n}) .

Therefore, specifying the images of the basis vectors determines the whole linear map, and in the standard basis those images become the columns of the matrix.