linear algebra線型代数せんけいだいすう portal

1Introduction

This page is the entrance to the foundational linear algebra線型代数せんけいだいすう lectures. The first goal is to understand what linearity線型性せんけいせい means and what the columns列れつ of a matrix行列ぎょうれつ record. From there the course connects systems of linear equations連立一次方程式れんりついちじほうていしき, rank階数かいすう, inner products内積ないせき, and eigenvalues固有値こゆうち.

The English route should be read as the same course map, not as a shortened summary. Start with linearity線型性せんけいせい and linear maps線型写像せんけいしゃぞう, then connect vectors, matrices, row operations, rank, determinants, inverse matrices, inner products, eigenvalues, and the advanced topics. Each later topic reuses earlier invariants: rank measures dimension, determinants detect invertibility, inner products define orthogonality, and eigenvectors expose directions preserved by a linear map.

When choosing a next page, check what kind of question you are answering. Computational questions usually point to row reduction, determinants, inverse matrices, or matrix products. Structural questions usually point to bases, kernels, images, rank, similarity, or diagonalization. Metric questions use inner products, projections, least squares, orthogonal diagonalization, SVD, and pseudoinverses.

2Overall map

The central object is not a table of numbers but a linear map線型写像せんけいしゃぞう: a map that preserves sums和わ and scalar multiplicationスカラー倍. Because of linearity線型性せんけいせい, once the images of the basis vectors基底ベクトル are known, the image of every vector is determined. A matrix行列ぎょうれつ records those images as its columns.

With this viewpoint, a matrix product行列積ぎょうれつせき represents composition合成ごうせい of maps, rank階数かいすう is the dimension次元じげん of the image, and the kernel核かく consists of input directions that collapse to zero. Adding an inner product内積ないせき introduces length, angle, orthogonality, and projection. Eigenvalues固有値こゆうち and diagonalization対角化たいかくか let us read a linear map線型写像せんけいしゃぞう by directions that are only scaled.

3Assumptions and scope

The basic setting is a finite-dimensional vector spaceベクトル空間 over the real or complex numbers. For real matrices the transpose is written $A^T$A^T; for complex matrices the conjugate transpose共役転置きょうやくてんち is written $A^{*}$A^*. Claims that require a square matrix正方行列せいほうぎょうれつ are kept separate from claims that also make sense for rectangular matrices長方行列ちょうほうぎょうれつ.

The important habit is to keep the hypotheses of every theorem. Orthogonal diagonalization of a symmetric matrix does not apply to an arbitrary diagonalizable matrix. Tests using a quadratic form二次形式にじけいしき require the relevant stationary-point condition. Later topics remain rigorous only when these conditions are kept visible.

4Intuitive guide

The intuition of linear algebra線型代数せんけいだいすう is easier to unify when a matrix行列ぎょうれつ is viewed not as an array of numbers but as a linear transformation線型変換せんけいへんかん that moves space.

Keeping this overview first connects elimination掃はき出だし, determinants行列式ぎょうれつしき, diagonalization対角化たいかくか, and orthogonalization直交化ちょっこうか not as separate computational procedures but as one series of operations about how space is preserved, where it collapses, and which directions become easier to see.

5What changes and what is preserved

In linear algebra線型代数せんけいだいすう, it is important to track not only the calculation itself but also what each operation操作そうさ changes and what it leaves unchanged. Even when the displayed form of the same matrix行列ぎょうれつ changes, the map itself or the solution set may remain unchanged. Conversely, rank階数かいすう may remain unchanged while the meaning of the unknowns changes.

This table previews properties proved in later lectures. Although it looks slightly ahead in the order of study, the reason for considering each operation操作そうさ becomes easier to understand once the preserved quantities are visible.

60. Linearity, linear maps, and matrix columns

Linearity線型性せんけいせい means that applying the map after forming a linear combination線型結合せんけいけつごう gives the same result as first applying the map and then using the same coefficients:

Therefore the images of basis基底きてい vectors determine the whole map. The $j$j-th column of a matrix is the coordinate vector of the image of the $j$j-th basis基底きてい vector.

71. Basic computational tools

After adopting the viewpoint of linear maps線型写像せんけいしゃぞう, the next step is to prepare the computational tools needed to handle matrices行列ぎょうれつ concretely. Matrix addition, scalar multiplication, matrix multiplication, the identity matrix単位行列たんいぎょうれつ, the zero matrix, and transpose are the basic vocabulary for working with linear maps in coordinates.

The caution here is not to detach the computation from the linear map. Matrix multiplication represents composition合成ごうせい of maps, the identity matrix単位行列たんいぎょうれつ represents the map that changes nothing, and the zero matrix represents the map that sends every input to zero. Keeping this meaning while entering component calculations makes the rules feel like structure rather than memorized formulas.

82. Systems, elementary operations, and invertibility

The equation $Ax=b$Ax=b asks for an input $x$x that reaches the output $b$b. Gaussian elimination掃はき出だし法ほう is both a solution method and a way to detect rank階数かいすう, image像ぞう, kernel核かく, and degrees of freedom. Row operations preserve equivalence of equations. Column operations preserve column space列空間れつくうかん and rank階数かいすう but change the coordinates of unknowns. An inverse matrix逆行列ぎゃくぎょうれつ is understood as the condition that no information is lost and every output has a unique recoverable input.

93. Change of basis基底変換きていへんかん and similarity相似そうじ

A change of basis changes the coordinate representation, not the vector or linear map線型写像せんけいしゃぞう itself. Similarity is the relation that represents the same linear map線型写像せんけいしゃぞう in another basis, and it prepares for diagonalization.

104. Determinants行列式ぎょうれつしき, volume scaling, and invertibility可逆性かぎゃくせい

A determinant行列式ぎょうれつしき measures how a square matrix正方行列せいほうぎょうれつ scales area or volume and whether orientation is preserved or reversed. A nonzero determinant means space is not collapsed and the input can be recovered.

115. Inner products, orthogonality, and projection

Linearity alone does not measure length or angle. Adding an inner product introduces length, angle, orthogonality, and projection. orthogonal projection直交射影ちょっこうしゃえい chooses the nearest point in a subspace部分空間ぶぶんくうかん and gives the geometric reason for least squares.

126. eigenvalues固有値こゆうち, eigenvectors固有こゆうベクトル, and diagonalization

An eigenvalue固有値こゆうち is the scale factor along an eigenvector固有こゆうベクトル direction whose line is not changed by the transformation. Diagonalization chooses a basis基底きてい of eigenvectors固有こゆうベクトル so that the linear map線型写像せんけいしゃぞう can be read direction by direction.

137. Where inner products and eigenvalues固有値こゆうち meet

For symmetric and Hermitian matrices, eigenvectors固有こゆうベクトル can be chosen as an orthonormal basis基底きてい. Quadratic forms and positive definite matrices connect eigenvalues固有値こゆうち to signs, convexity, and optimization.

Over the complex numbers, normal matrices正規行列せいきぎょうれつ are the natural criterion for unitary diagonalization. A Hermitian matrix is a special normal matrix whose eigenvalues固有値こゆうち are real, so it connects smoothly to quadratic forms and positive definiteness.

In the complex route, it is easier to proceed from conjugate transpose共役転置きょうやくてんち and adjoints to Hermitian matrices, unitary matrices, and normal matrices.

148. Entrance to advanced topics

Singular value decomposition特異値分解とくいちぶんかい and the pseudoinverse擬似逆行列ぎじぎゃくぎょうれつ connect rectangular or rank階数かいすう-deficient matrices to projection and least squares. The minimal polynomial最小多項式さいしょうたこうしき and the companion matrix同伴行列どうはんぎょうれつ connect polynomials with matrices, while Jordan normal formジョルダン標準形ひょうじゅんけい describes matrices that cannot be diagonalized.

15Exercise links

After checking definitions and reasons in the lectures, the exercises ask not only whether a computation can be done but also what each operation preserves and what it changes. Worked examples are used to stabilize understanding, while exercise problems check whether you can make the judgment yourself.

16Recommended order

16.1Recommended route starting from linearity線型性せんけいせい

1. Check linearity線型性せんけいせい through additivity and homogeneity.
2. Check the basic operations on vectorsベクトル as operations preserved by linearity.
3. Check linear combinations線型結合せんけいけつごう and spans.
4. Check that a basis基底きてい represents every vector uniquely.
5. Understand a linear map線型写像せんけいしゃぞう as a transformation determined by the images of basis vectors.
6. Read the columns列れつ of a matrix as images of basis vectors.
7. Read matrix products as composition of linear maps.
8. Read rank階数かいすう as image dimension and kernel核かく as input directions collapsed to zero.
9. Interpret systems of linear equations, row operations, and invertibility as information preservation.
10. Use change of basis to develop the habit of changing only the representation.
11. Connect determinants to volume scale factors, inner products to length and angle, and eigenvalues固有値こゆうち to direction-wise scale factors.

16.2Computation-first auxiliary route

1. Check matrix operations and size conditions.
2. Use elementary row operations and echelon forms to proceed toward Gaussian elimination.
3. Check inverse-matrix procedures and determinant calculation rules.
4. Reinterpret pivots, rank階数かいすう, and determinants as the image, kernel, and volume scale factor of a linear map線型写像せんけいしゃぞう.