찾아보기찾아보기

$\cB$ -coordinates

change of basis matrix중요

computing

row reduction중요

with respect to an orthogonal basis중요

definition of정의

informally문단

labeling points중요

nonstandard grid예제

Algebraic multiplicity

and diagonalizability정리

and geometric multiplicity정리

definition of정의

equals one중요

of similar matrices정리

Approximate solution참조 Least-squares

Augmented matrix참조 Matrix

Basis

and orthogonal projection따름정리

basis theorem기저 정리

coordinates with respect to참조 $\cB$ -coordinates

definition of정의

infinitely many중요

making orthogonal정리

of $\R^n$ 예제 예제

of a column space정리

of a null space정리

of a span문단

span of an orthogonal set사실

uniqueness with respect to사실

Best-fit problem소절

best-fit ellipse예제

best-fit line예제

best-fit linear equation예제

best-fit parabola예제

best-fit trigonometric function예제

general setup문단

Block Diagonalization Theorem블록 대각화 정리

Characteristic polynomial

and eigenvalues정리

definition of정의

factoring by hand문단

form of정리

of a $2\times 2$ matrix중요

of similar matrices사실

Codomain참조 Transformation

Cofactor참조 Matrix

Color space예제

Column rank참조 Rank

Column space

and invertibility가역행렬 정리

and rank참조 Rank

basis of참조 Basis

definition of정의

is a subspace문단

is row space of transpose문단

of an orthogonal projection명제

orthogonal complement of명제 중요

range of a transformation중요

versus the solution set소절

Column span참조 Column space

Complex conjugation참조 Complex numbers

Complex eigenvalue

$2\times 2$ matrices

and rotation-scaling matrices회전-확대 정리

and rotation-scaling matrices, computing중요

different rotation-scaling matrices문단

dynamics of노트 예제 예제 예제

geometry of소절

$3\times 3$ matrices

dynamics of예제

conjugate pairs중요

existence of중요

Complex numbers

absolute value항목

arithmetic of노트

conjugation항목

definition of정의

real and imaginary parts of항목

vectors문단

Consistent참조 System of linear equations

Cramer’s rule정리

and computing inverses정리

Determinant

alternative defining properties of참고

and column operations따름정리

and computing inverses정리

and powers of matrices따름정리

and row operations정의

and volumes정리

computation of

cofactor expansion정리

row reduction계산 절차

defining properties of정의

existence and uniqueness of정리

identity matrix정의

invertibility property명제 가역행렬 정리

methods of computation중요

multilinearity property명제

multiplicativity property명제

and volumes참고

of a $2\times 2$ matrix정의 예제 예제

of a $3\times 3$ matrix중요

of similar matrices문단

properties of중요

transpose property명제

Diagonal

see Matrix정의

Diagonalizability

algebraic-geometric multiplicity criterion정리

criterion대각화 정리

criterion for similarity중요

definition of정의

diagonal matrices예제 예제

distinct eigenvalues중요

dynamics of예제 예제 예제 예제

geometry of소절

is unrelated to invertibility중요

of $2\times 2$ matrices예제

of $3\times 3$ matrices예제

of a projection matrix명제

order of eigenvalues노트

powers of중요

projection예제

recipe중요

shear예제

similar matrices예제

Difference equation정의

Dilation예제 예제 예제

eigenvectors of예제

inverse of예제 예제

Dimension

definition of정의

of a column space정의

of a null space정의

of a solution set중요 중요

of an orthogonal complement사실

Domain참조 Transformation

Dot product

and angles참고

and distance정의

and length사실

definition of정의

properties of노트

Eigenspace

and the null space사실

computation중요

definition of정의

is a solution set문단

is a subspace문단

of a projection matrix명제

of similar matrices중요

orthogonal complement of예제

Eigenvalue

algebraic multiplicity of참조 Algebraic multiplicity

and diagonalizability중요

and invertibility가역행렬 정리

and stochastic matrices사실

and the characteristic polynomial참조 Characteristic polynomial

complex참조 Complex eigenvalue

definition of정의

eigenvector for정의

geometric multiplicity of참조 Geometric multiplicity

identity matrix예제

maximum number of중요

of a projection matrix명제

of a triangular matrix따름정리

of similar matrices중요

zero사실

Eigenvector

and collinearity문단

and diagonalizability대각화 정리

and stochastic matrices정의

computation중요

trick for $2\times 2$ matrices노트

definition of정의

eigenvalue for정의

identity matrix예제

linear independence of사실

of a projection matrix명제

of similar matrices사실

Elimination method소절

Ellipse

area of참조 Volume

best-fit problem예제

Equation of linear dependence참조 Linear independence

Free variable정의

Function참조 Transformation

Fundamental theorem of algebra대수학의 기본 정리

Gaussian elimination참조 Row reduction

Geometric multiplicity

and algebraic multiplicity정리

and diagonalizability정리

definition of정의

of similar matrices정리

Google PageRank

Google Matrix정의

eigenvector of중요

importance matrix정의

eigenvector of중요

importance rule정의

Gram–Schmidt Process정리

detecting linear dependence중요

Homogeneous참조 System of linear equations

Identity matrix

and identity transformation예제

and matrix multiplication노트

and standard coordinate vectors정의

as a function예제

as a matrix transformation예제

definition of정의

determinant of정의

eigenvalues of예제

eigenvectors of예제

similarity문단

Identity transformation

and composition노트

and identity matrix예제

definition of정의

Imaginary number참조 Complex numbers

Imaginary part참조 Complex numbers

Implicit equation소절 노트

Inconsistent참조 System of linear equations

Increasing span criterion참조 Linear independence

Inhomogeneous참조 System of linear equations

Invertible matrix

and invertible transformation정리

basic facts사실

computation

$2\times 2$ case명제

in general정리

using Cramer’ rule정리

definition of정의

determinant of명제 가역행렬 정리

inverse of항목

invertible matrix theorem가역행렬 정리

solving linear systems with정리

Invertible matrix theorem가역행렬 정리 가역행렬 정리

Invertible transformation

and invertible matrices정리 가역행렬 정리

definition of정의

one-to-one and onto명제

Least-squares

and $Ax=b_{\Col(A)}$ 중요

computation of

complicated matrix formula정리

Projection Formula중요

row reduction정리 중요

definition of정의

picture of문단

uniqueness of정리

Line

dimension-1 solution set중요 중요

geometric definition of정의

number line예제

orthogonal projection onto중요 예제

parametric form of예제

Linear combination

collinear vectors, picture of예제

definition of정의

single vector, picture of예제

two vectors, picture of예제

Linear dependence참조 Linear independence

Linear dependence relation참조 Linear independence

Linear equation

definition of정의

system of참조 System of linear equations

Linear independence

and determinants따름정리

and invertibility가역행렬 정리

basic facts사실

definition of정의

equation of linear dependence문단

increasing span criterion정리

linear dependence relation문단

of an orthogonal set사실

pictures of소절

verifying중요

with Gram–Schmidt중요

wide matrices중요

Linear transformation

addition of참조 Transformation

and volumes참조 Matrix transformation

are matrix transformations중요

basic facts사실

composition of참조 Transformation

and matrix multiplication정리

linearity of정리

definition of정의

dictionary노트

invertible참조 Invertible transformation

scalar multiplication of참조 Transformation

standard matrix of정리

orthogonal projection문단

verifying nonlinearity예제

when defined by a formula문단

Lower-triangular

see Matrix정의

Matrix

addition of정의

as a function참조 Matrix transformation문단

augmented문단

cofactor of정의

and determinants정리

sign of문단

definition of문단

determinant of참조 Determinant

diagonal entries of정의

inverse of참조 Invertible matrix

invertible참조 Invertible matrix

lower-triangular정의

determinant of명제

eigenvalues of따름정리

minor of정의

multiplication참조 Matrix multiplication

nullity of참조 Nullity

parallelepiped determined by정리

product with vector참조 Matrix-vector product

projection참조 Orthogonal projection, standard matrix of

rank of참조 Rank

rotation-scaling참조 Rotation-scaling matrix

scalar multiplication of정의

similar참조 Similarity

size of중요

stochastic참조 Stochastic matrix

trace of정의

similar matrices문단

transpose of정의

and products사실

determinant of명제

upper-triangular정의

determinant of명제

eigenvalues of따름정리

Matrix equation

always consistent정리

and invertibility가역행렬 정리

definition of정의

equivalence with vector equation노트

solving with the inverse matrix정리

spans and consistency참고

Matrix multiplication

and composition of transformations정리

and the matrix-vector product문단

associativity of노트

caveats중요

definition of정의

determinant of명제

inverse of항목

noncommutativity of예제

order of operations예제

powers문단

and diagonalizability중요

and similarity사실

properties of노트

row-column rule중요

size of matrices중요

Matrix transformation

addition of참조 Transformation

and volumes정리

codomain of중요

composition of참조 Transformation참조 Linear transformation

definition of정의

dictionary노트

domain of중요

invertible참조 Invertible transformation

linearity of문단

of $\R^2$ 예제

one-to-one criteria정리

onto criteria정리

range of중요

scalar multiplication of참조 Transformation

tall matrices노트

wide matrices노트

Matrix-vector product

and matrix multiplication문단

definition of정의

row-column rule중요

with standard coordinate vectors사실

Minor참조 Matrix

Multiplicity

algebraic참조 Algebraic multiplicity

geometric참조 Geometric multiplicity

Nontrivial solution참조 System of linear equations

Null space

and invertibility가역행렬 정리

basis of참조 Basis

computing중요

definition of정의

is a solution set예제

is a subspace소절

is the $0$ -eigenspace사실

of an orthogonal projection명제

orthogonal complement of중요

Nullity정의

rank theorem랭크 정리

One-to-one

and invertibility가역행렬 정리

criteria for matrix transformations정리

definition of정의

equivalent formulations문단

finding two vectors with the same image문단

functions of one variable예제

negation of문단

square matrices노트

versus onto문단

wide matrices노트

Onto

and invertibility가역행렬 정리

criteria for matrix transformations정리

definition of정의

equivalent formulations문단

finding a vector not in the range문단

functions of one variable예제

negation of문단

square matrices노트

tall matrices노트

versus one-to-one문단

Orthogonal complement

basic facts사실

computation of소절

definition of정의

dimension of사실

of a column space명제 중요

of a null space중요

of a row space중요

of a span문단 중요

of an eigenspace예제

orthogonal complement of사실

pictures of문단

system of linear equations문단

Orthogonal decomposition참조 Orthogonal projection

Orthogonal projection

and $\cB$ -coordinates중요

as a transformation소절

composed with itself명제

computation of

complicated matrix formula따름정리

Projection Formula사영 공식

row reduction정리 중요

definition of정의

distance from중요

eigenvectors of예제

existence of정리

is the closest vector중요

linearity of명제

of a vector in $W$ 예제

of a vector in $W^\perp$ 예제

onto a column space중요

onto a line중요 예제

onto the $xy$ -plane예제 예제 예제

noninvertibility of비예제

properties of명제

range of명제

standard matrix of문단

column space of명제

complicated matrix formula중요

diagonalizability of명제

eigenvalues of명제

eigenvectors of명제

noninvertibility of명제

null space of명제

properties of명제

square of명제

Orthogonal set

and $\cB$ -coordinates중요

and least squares중요

definition of정의

linear independence of사실

making orthonormal중요

necessity of중요

orthonormality of정의

standard coordinate vectors예제

producing from a basis정리

Orthogonality

and the Pythagorean theorem중요

definition of정의

zero vector중요

Orthonormal set참조 Orthogonal set

Parallelepiped

definition of정의

flat핵심 관찰

parallelogram예제

area of예제

volume of정리

Parallelogram참조 Parallelepiped

Parameterized equation노트

Parametric form소절 중요

Parametric vector form

of a homogeneous equation예제 중요

of an inhomogeneous equation문단

particular solution핵심 관찰

Particular solution참조 Parametric vector form

Perron–Frobenius theorem정리

Pivot정의

Pivot position정의

Plane

dimension-2 solution set중요 중요

geometric definition of정의

parametric form of예제

$xy$ -plane예제

Point정의

distance between정의

Polynomial

characteristic참조 Characteristic polynomial

complex roots대수학의 기본 정리

conjugate roots참고

cubic예제

factoring by hand문단

quadratic예제

rational roots유리근 정리

Power of a matrix참조 Matrix multiplication

Projection참조 Orthogonal projection

diagonalizability of예제

Projection Formula사영 공식

Projection matrix참조 Orthogonal projection, standard matrix of

QR codes예제

Quadratic formula예제

Range참조 Transformation

Rank정의

rank theorem랭크 정리

row and column소절

Rational Root Theorem유리근 정리

Real $n$ -space정의

as a subspace of itself예제

point of정의

Real numbers $\R$ 문단 예제

Real part참조 Complex numbers

Red Box예제 예제 예제

Reduced row echelon form정의

and invertibility가역행렬 정리

Reflection

eigenspaces of예제

eigenvectors of예제

in general참고

over the $y$ -axis예제 예제

inverse of예제 예제

Rotation

composition of예제

counterclockwise by $45^\circ$ 예제 예제

counterclockwise by $90^\circ$ 예제 예제

counterclockwise by $\theta$ 예제

eigenvectors of예제

non-diagonalizability of예제

Rotation-scaling matrix

and complex eigenvalues회전-확대 정리

computing the angle중요

definition of정의

structure of명제

Rotation-Scaling Theorem회전-확대 정리

Row echelon form정의

Row equivalence정의

Row operations문단

and determinants정의

replacement항목

scaling항목

swap항목

Row rank참조 Rank

Row reduction

algorithm알고리즘

computing determinants계산 절차

picture of문단

Row replacement참조 Row operations, replacement

Row space

definition of정의

is column space of transpose문단

orthogonal complement of중요

Row vector참조 Vector

Scale참조 Dilation참조 Row operations, scaling

Set builder notation노트

Shear

eigenvectors of예제

in the $x$ -direction예제 예제

multiplicities예제

non-diagonalizability of예제

Similarity

action on a vector중요

and eigenspaces중요

and eigenvalues중요

and eigenvectors사실

and multiplicities정리

and powers사실

and the characteristic polynomial사실

and the determinant문단

and the trace문단

definition of정의

equivalence relation명제

geometry of소절

identity matrix문단

of $2\times 2$ matrices참고

of diagonal matrices예제

of diagonalizable matrices중요

to a diagonal matrix참조 Diagonalizability

worked example예제

Solution참조 System of linear equations

Solution set

definition of정의

of a homogeneous system is a null space예제

of a homogeneous system is a span중요

picture of소절

size of소절

translate of a span핵심 관찰

versus the column space소절

Space

$\R^3$ 예제

color space예제

dimension-3 solution set중요 중요

Span

basis of참조 Basis

definition of정의

is a subspace정리

orthogonal complement of문단 중요

pictures of문단

Standard coordinate vectors

and matrix columns사실

are unit vectors문단

columns of the identity matrix정의

definition of표기법

orthonormality of예제

picture of문단

Standard matrix참조 Linear transformation

Steady state참조 Stochastic matrix

Stochastic matrix

definition of정의

eigenvalues of사실

long-term behavior of정리

picture of그림 그림

steady state of정의

computing중요 중요

sum of entries of vector중요

Subset

definition of정의

set builder notation노트

versus subspace중요

Subspace

and spans정리

definition of정의

is a span정리

orthogonal complement of참조 Orthogonal complement

real $n$ -space예제

versus subset중요

writing as column or null space문단

zero예제

Superposition principle문단

System of linear equations

consistent정의

picture of노트

span criterion노트 참고

definition of문단

four ways of writing노트

homogeneous정의

trivial solution중요

inconsistent정의

picture of그림

RREF criterion정리

inhomogeneous정의

nontrivial solution중요

and free variables관찰

and invertibility가역행렬 정리

finding중요

number of solutions of소절

parametric form of참조 Parametric form

parametric vector form of참조 Parametric vector form

particular solution of참조 Parametric vector form

solution of정의

solving with the inverse matrix정리

trivial solution중요

and invertibility가역행렬 정리

Tall matrix참조 Matrix transformation

Trace참조 Matrix

Transformation

addition of정의

as a machine문단

associated to a matrix참조 Matrix transformation

codomain of정의

composition of정의

noncommutativity of중요

order of operations예제

definition of정의

domain of정의

identity참조 Identity transformation

invertible참조 Invertible transformation

linear참조 Linear transformation

of one variable예제

of several variables예제

one-to-one참조 One-to-one

onto참조 Onto

range of정의

scalar multiplication of정의

Transpose참조 Matrix

Triangle

area of참조 Volume

Trivial solution참조 System of linear equations

Unit cube예제

Unit vector

and orthonormality정의

definition of정의

in the direction of a vector사실

standard coordinate vectors문단

Upper-triangular

see Matrix정의

Vector

addition항목

parallelogram law문단

angle between참고

definition of정의

distance between정의

length of사실

linear combination of참조 Linear combination

orthogonal참조 Orthogonality

product with matrix참조 Matrix-vector product

real and imaginary parts of문단

row vector정의

product with column vector정의

scalar multiplication항목

picture of문단

subtraction

picture of문단

unit vector참조 Unit vector

unit vector in the direction of사실

Vector equation

consistent참조 System of linear equations, consistent

definition of정의

equivalence with matrix equation노트

equivalence with system of equations문단

inconsistent참조 System of linear equations, inconsistent

solving중요

Volume

and length예제

of a parallelepiped정리

of a region정리

of a triangle예제

of an ellipse예제

signed참고

Wide matrix참조 Matrix transformation참조 Linear independence

번역에 대한 의견이나 오류 제보(GitHub 계정 필요)