Linear maps線型写像せんけいしゃぞう and matrices行列ぎょうれつ

1Introduction

In linear algebra線型代数せんけいだいすう, the most important point is to understand a matrix行列ぎょうれつ not merely as a table of numbers, but as a coordinate representation座標表示ざひょうひょうじ of how a map moves vectors.

If one follows only matrix calculations, the rules of multiplication and component calculation remain, but the main object is the action作用さよう on a space空間くうかん. This lecture connects two facts: a linear map線型写像せんけいしゃぞう is determined by the images of a basis基底きてい, and each column of a matrix records those images in coordinates.

2Terms and definitions

A linear map線型写像せんけいしゃぞう is a map写像しゃぞう that preserves vector addition加法かほう and scalar multiplicationスカラー倍ばい. Precisely, for all vectors $u,v$u,v and all scalars $c$c,

must hold.

The first condition,

is called additivity加法性かほうせい. It says that adding first and then mapping gives the same result as mapping first and then adding. Geometrically, the parallelogram rule for vector addition is not broken.

The second condition,

is called homogeneity同次性どうじせい. It says that scaling first and then mapping gives the same result as mapping first and then scaling. Geometrically, scale relationships along lines through the origin are preserved.

Linearity線型性せんけいせい means satisfying additivity and homogeneity at the same time. One condition alone is not enough. More generally, a linear map線型写像せんけいしゃぞう satisfies

Thus forming a linear combination線型結合せんけいけつごう before applying the map is the same as applying the map to each vector and then taking the same linear combination.

This property is why the images of the basis vectors determine the whole map. Without linearity線型性せんけいせい, knowing the images of the basis vectors would not determine where their sums or scalar multiples go.

A linear map線型写像せんけいしゃぞう preserves sums, scalar multiples, linear combinations, and the structure of subspaces部分空間ぶぶんくうかん. It does not necessarily preserve everything: different vectors may collapse to the same vector, and lengths, angles, areas, and volumes may change. A useful interpretation is that a linear map preserves the additive and scaling skeleton of a vector space.

A matrix行列ぎょうれつ is a representation表現ひょうげん of a linear map after bases have been chosen.

A coordinate map座標写像ざひょうしゃぞう sends a vector $v$v to its coordinate vector座標ざひょうベクトル $[v{]}_B$[v]_B with respect to a basis $B$B.

A representation matrix表現行列ひょうげんぎょうれつ of a linear map線型写像せんけいしゃぞう $T:V\to W$T:V\to W is the matrix that describes $T$T in coordinates with respect to a basis $B$B of the domain and a basis $C$C of the codomain.

3Strategy

To understand a linear map線型写像せんけいしゃぞう, first check where each basis vector goes. Then linearity線型性せんけいせい determines the entire map.

4Intuitive explanation

4.11. Interpretation as a map

A linear map線型写像せんけいしゃぞう is a transformation that sends the origin原点げんてん to the origin and preserves sums and scalar multiples. A line through the origin is sent to a line through the origin or collapses to a point. Expansions, contractions, rotations, and shears are typical examples of linear maps.

The reason for this definition is that it extracts transformations that do not break the basic operations of vector spaces: vector addition and scalar multiplication. This is what makes the whole map recoverable from the images of a basis.

4.22. Meaning of column vectors

If the columns of a matrix行列ぎょうれつ are treated as mere lists of numbers, their meaning is hidden. In a general basis, each column is not necessarily the image vector itself; it is the coordinate vector of the image of one basis vector in the codomain basis. Only for the standard basis標準基底ひょうじゅんきてい can the column be read directly as the image vector itself.

4.33. Connection with calculation

When a matrix is multiplied by a coordinate vector, the coefficients of the input vector are used to take a linear combination線型結合せんけいけつごう of the images of the basis vectors. Matrix multiplication is therefore not a mysterious rule. It is the operation of constructing the output by combining the images of the basis vectors.

5Precise explanation

5.11. Images of a basis determine the map

Choose a basis基底きてい $e_1,\text{[PARSE ERROR: Undefined("Command("dots")")]},e_n$e_1,\dots,e_n. Every vector $v$v can be written as

By linearity線型性せんけいせい,

Therefore, once $T\left(e_i\right)$T(e_i) is known for every basis vector, $T$T is completely determined. This is the precise form of the statement that the columns of a matrix record the images of basis vectors.

5.22. What the columns of a matrix are

First consider the standard basis標準基底ひょうじゅんきてい. The matrix of a linear map is obtained by arranging the vectors $T\left(e_i\right)$T(e_i) as columns:

For an input vector $x=(x_1,\text{[PARSE ERROR: Undefined("Command("dots")")]},x_n{)}^T$x=(x_1,\dots,x_n)^T,

So matrix-vector multiplication is a column-wise linear combination.

5.33. Coordinate maps and representation matrices

Let the ordered basis of the domain $V$V be

and let the ordered basis of the codomain $W$W be

The coordinate maps are

The representation matrix表現行列ひょうげんぎょうれつ is the matrix $[T{]}_{C\leftarrow B}$[T]_{C\leftarrow B} satisfying

If

then the coefficients $a_{ij}$a_{ij} form the matrix

Equivalently, the $j$j-th column is

5.44. Composition and matrix products

For linear maps

with compatible bases $B,C,D$B,C,D,

Thus a matrix product represents composition of maps. In $ABx$ABx, the matrix $B$B acts first and $A$A acts second.

5.55. Concrete example with the standard basis

Suppose

Then the matrix is

For an input vector $(x,y{)}^T$(x,y)^T,

This example shows directly that each column stores the action of the map on one basis vector.

5.66. Concrete example with a nonstandard basis

The same map can have a different representation matrix when the basis is changed. Consider the linear map on $R^2$\mathbb R^2

In the standard basis,

Now use the nonstandard basis非標準基底ひひょうじゅんきてい

Then

and

Therefore the representation matrix with respect to $B$B is

Check this with the concrete vector $x=(3,1{)}^T$x=(3,1)^T. In the basis $B$B,

Then

This coordinate vector represents

Directly applying the map also gives

The map did not change; the coordinate system and the representation matrix changed.

5.77. How a representation matrix changes with a basis

Order note: this subsection previews later lectures on change of basis基底変換きていへんかん, similarity相似そうじ, and eigenvalues固有値こゆうち. Here $$P^{-1}$$P^{-1} means the inverse matrix逆行列ぎゃくぎょうれつ satisfying $$P^{-1}P=P{P}^{-1}=I$$P^{-1}P=PP^{-1}=I, and similarity相似そうじ means the relation $$A_{\mathrm{n}\mathrm{e}\mathrm{w}}=P^{-1}{A}_{\mathrm{o}\mathrm{l}\mathrm{d}}P$$A_{\mathrm{new}}=P^{-1}A_{\mathrm{old}}P. The proofs that eigenvalues, characteristic polynomials, and minimal polynomials are preserved by similarity are deferred to later lectures.

For the same vector space, suppose the old and new coordinates are related by the change-of-coordinates matrix座標変換行列ざひょうへんかんぎょうれつ $P$P:

If

then converting the output back to new coordinates gives

Therefore

If the bases in the domain and codomain are changed separately, using $P$P on the domain side and $Q$Q on the codomain side gives

When only the basis of the same space is changed, matrices related by $A_{\mathrm{n}\mathrm{e}\mathrm{w}}=P^{-1}{A}_{\mathrm{o}\mathrm{l}\mathrm{d}}P$A_{\mathrm{new}}=P^{-1}A_{\mathrm{old}}P are called similar相似そうじ. Similar matrices represent the same linear map線型写像せんけいしゃぞう in different bases and share invariants such as eigenvalues, characteristic polynomial, and minimal polynomial.

6Bridge to later concepts

This section previews how elementary column operations, rank, and determinants are read from the linear-map viewpoint. Rank and determinants are defined in detail later, but the structural interpretation starts here.

6.11. The column space is the image

Let $A=[a_1\cdots a_n]$A=[a_1\ \cdots\ a_n] and define $T_A:K^n\to K^m$T_A:K^n\to K^m by $T_A\left(x\right)=Ax$T_A(x)=Ax. Then

Indeed, for $x=(x_1,\dots ,x_n{)}^T$x=(x_1,\ldots,x_n)^T,

Thus every output is a linear combination of the columns, and every linear combination of the columns is obtained as $Ax$Ax for a suitable $x$x.

6.22. Column operations are invertible input-side changes

If $F$F is invertible, then $AF$AF represents the composite map $T_A\circ T_F$T_A\circ T_F, because

Since $F$F maps $K^n$K^n onto $K^n$K^n,

Therefore

so rank is preserved.

6.33. The determinant is the square case volume scale factor

For a square matrix, the determinant, introduced later, measures oriented volume scaling. If rank is less than $n$n, $n$n-dimensional volume collapses and the determinant is zero. Later this becomes the theorem

7Other viewpoints

7.1Geometric viewpoint

A linear map線型写像せんけいしゃぞう stretches, contracts, rotates, or shears a plane or space while preserving the linear structure. From this viewpoint, the image像ぞう records where vectors can be sent, and the kernel核かく records which directions collapse to zero.

Solving $Ax=0$Ax=0 for a matrix $A$A detects directions that the map sends to the origin. Geometrically, Gaussian elimination for finding the kernel checks which degrees of freedom disappear.

7.2Algebraic viewpoint

From the perspective of coordinate calculation, a matrix is a table of components. By tracking how the input coefficients are recombined, one can read the action of the map as a formula.

7.3Structural viewpoint

From the perspective of maps, a matrix represents how a space is transformed. The bridge between the computational and structural viewpoints is the image of a basis. In university mathematics, it is natural to regard the matrix not as the map itself, but as the expression of a linear map線型写像せんけいしゃぞう after a basis has been chosen.

8Scope of validity

A matrix行列ぎょうれつ depends on the chosen basis基底きてい, but the linear map線型写像せんけいしゃぞう itself does not.

Changing the basis can turn the same map into a different matrix. Therefore, if one looks only at the matrix and thinks it is the transformation itself, the basis-dependence of the representation is missed.

9Criteria

• If the meaning of each column is being asked, interpret it as the image of a basis vector, or as the coordinates of that image in the codomain basis.
• If the meaning of a matrix product becomes unclear, reduce it to a linear combination of column vectors.
• If a basis change appears, distinguish the underlying map from its matrix representation.

10Final forms

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{Alinearmapisdeterminedbytheimagesofabasis}$\boxed{\text{A linear map is determined by the images of a basis}}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{Amatrixisacoordinaterepresentationofalinearmap}$\boxed{\text{A matrix is a coordinate representation of a linear map}}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\left[T\right(x){]}_C=[T{]}_{C\leftarrow B}[x{]}_B$\boxed{[T(x)]_C=[T]_{C\leftarrow B}[x]_B}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{column}j=\left[T\right(v_j){]}_C$\boxed{\text{column }j=[T(v_j)]_C}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\left[T\right(v){]}_C=A[v{]}_B$\boxed{[T(v)]_C=A[v]_B}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}[S\circ T{]}_{D\leftarrow B}=[S{]}_{D\leftarrow C}[T{]}_{C\leftarrow B}$\boxed{[S\circ T]_{D\leftarrow B}=[S]_{D\leftarrow C}[T]_{C\leftarrow B}}

$\text{[PARSE ERROR: Undefined("Command("boxed")")]}\text{Basischangeandsimilarityaretreatedindetailonalaterpage}$\boxed{\text{Basis change and similarity are treated in detail on a later page}}

11In one sentence

• The actual object behind a matrix行列ぎょうれつ is a linear map線型写像せんけいしゃぞう, and the matrix is its coordinate representation after a basis基底きてい has been chosen.
• In a general basis, a column records the coordinate vector of the image of a basis vector.
• When the basis changes, the same linear map線型写像せんけいしゃぞう can have a different representation matrix表現行列ひょうげんぎょうれつ.

This subsection is a preview of later lectures, not a primary definition for this lecture. It points ahead to change of basis and similarity, eigenvalues and eigenvectors, and minimal polynomials. Here we use only the minimum needed to see that a representation matrix depends on the chosen basis.

$P$P is the change-of-coordinates matrix座標変換行列ざひょうへんかんぎょうれつ that converts new coordinates into old coordinates. $P^{-1}$P^{-1} converts old coordinates back into new coordinates; its existence and computation belong to the later lectures on inverse matrices and change of basis. Matrices related by $A_{\mathrm{n}\mathrm{e}\mathrm{w}}=P^{-1}{A}_{\mathrm{o}\mathrm{l}\mathrm{d}}P$A_{\mathrm{new}}=P^{-1}A_{\mathrm{old}}P are called similar相似そうじ.

An eigenvalue固有値こゆうち is a scalar $\lambda$\lambda for which $Av=\lambda v$Av=\lambda v has some $v\ne 0$v\ne0. The characteristic polynomial特性多項式とくせいたこうしき is $\det (tI-A)$\det(tI-A), and the minimal polynomial最小多項式さいしょうたこうしき is the monic polynomial of least degree such that $m\left(A\right)=0$m(A)=0. These are treated later as quantities preserved by similarity.

A linear map線型写像せんけいしゃぞう is the coordinate-free object; a matrix行列ぎょうれつ is its coordinate representation after bases have been chosen. The columns of the representation matrix are the images of the domain basis vectors, written in the codomain basis, so the whole map is determined by what it does to a basis.

The main checks are therefore: preserve addition and scalar multiplication, read the kernel核かく as the set sent to $0$0, read the image像ぞう as the span of the columns, and compose maps in the same order as matrix multiplication permits. If the domain or codomain basis changes, the map has not changed, but the representing matrix has.

12Theorem and proof: images of a basis determine a linear map

Let $e_1,\text{[PARSE ERROR: Undefined("Command("dots")")]},e_n$e_1,\dots,e_n be a basis of $V$V. A linear map線型写像せんけいしゃぞう $T:V\to W$T:V\to W is uniquely determined by the vectors $T\left(e_1\right),\text{[PARSE ERROR: Undefined("Command("dots")")]},T\left(e_n\right)$T(e_1),\dots,T(e_n).

Indeed, every $v\in V$v\in V has a unique expression

By linearity,

Thus knowing the images of the basis vectors determines $T\left(v\right)$T(v) for every vector $v$v, and hence determines the whole map.

13Exercise links

14Related links