1.1: 线性方程组
Systems of Linear Equations

使用初等行运算解线性方程组
Use elementary row operations to solve systems of linear equations
判断线性方程组是否相容
Determine if a system of linear equations is consistent
确定线性系统相容的条件
Determine the conditions for which a linear system is consistent
判断关于线性方程组、行运算或矩阵的陈述的有效性
Determine the validity of statements about systems of linear equations, row operations, or matrices

1.2: 行简化与阶梯形态
Row Reduction and Echelon Forms

识别阶梯形矩阵和简化阶梯形矩阵
Identify matrices in echelon form and reduced echelon form
将矩阵行化简为简化阶梯形
Row reduce matrices to reduced echelon form
求给定增广矩阵方程组的通解
Find the general solution to a system with a given augmented matrix
根据对应系数矩阵的描述判断解是否一致
Determine if a solution is consistent given a description of the corresponding coefficient matrix
确定线性系统在何种条件下具有特定类型的解
Determine the conditions for which a linear system has specified types of solutions
判断关于行化简和阶梯形矩阵陈述的正确性
Determine the validity of statements about row reduction and echelon forms
运用行化简解决应用问题
Solve applications using row reduction

1.3: 向量方程
Vector Equations

计算向量的和与标量积，包括代数方法和几何方法
Compute sums and scalar products of vectors, both algebraically and geometrically
在向量方程与方程组之间进行转换
Convert between vector equations and systems of equations
判断一个向量是否为其他向量的线性组合
Determine if a vector is a linear combination of other vectors.
用代数或几何方法描述向量组的张成空间
Characterize the span of a set of vectors algebraically or geometrically.
判定关于向量及向量方程陈述的正确性
Determine the validity of statements about vectors and vector equations.
解决涉及向量方程的应用问题
Solve applications involving vector equations.

^HW1^

1.4: 矩阵方程 Ax=b
The Matrix Equation Ax = b

计算矩阵与向量的乘积
Compute the product of a matrix and a vector.
在矩阵方程、向量方程和方程组之间进行转换
Convert between matrix equations, vector equations, and systems of equations.
使用增广矩阵求解矩阵方程
Solve matrix equations using augmented matrices.
描述矩阵列向量的生成空间
Characterize the span of the column vectors of a matrix.
判断矩阵方程无解、有唯一解还是有无穷多解
Determine whether a matrix equation has no solution, one solution, or infinitely many solutions.
判定关于向量方程和矩阵方程的陈述是否成立
Determine the validity of statements about vector equations and matrix equations.
解决涉及矩阵方程的应用问题
Solve applications involving matrix equations.

1.5: 线性系统的解集
Solution Sets of Linear Systems

判断一个方程组是否有非平凡解
Determine if a system of equations has a nontrivial solution.
解方程组或矩阵方程，并以参数形式写出解
Solve a system of equations or a matrix equation and write the solution in parametric form.
从几何角度描述方程组的解集
Describe the solution sets of systems of equations geometrically.
判断关于线性方程组解集的陈述是否正确
Determine the validity of statements about solution sets of linear equations.

1.6*: 线性系统的应用
Applications of Linear Systems

^HW2^

1.7: 线性无关
Linear Independence

判断一组向量是否线性无关，并确定一个向量是否在给定的张成空间中
Determine if a set of vectors is linearly independent and determine if a vector is in a given span.
确定向量线性无关或具有给定张成空间的条件
Determine conditions for which vectors are linearly independent or have a given span.
判断关于线性独立性的陈述的有效性
Determine the validity of statements about linear independence.

^HW3^

– – -MID 1- – –

1.8: 线性变换简介
Introduction to Linear Transformations

$\displaystyle \mathbb{R}^n \rightarrow \mathbb{R}^m$

代数方法求解给定向量在线性变换下的图像
Algebraically find the image of a given vector under a linear transformation.
给定线性变换 $T(x)=A x$ ，针对像中的给定 $b$ 求解 $x$
Given a linear transformation T(x)=Ax, find x for a given b in the image of T.
确定线性变换具有给定定义域和陪域的条件
Determine the conditions for which a linear transformation has a given domain and codomain.

$\displaystyle \text { 当 } A \text { 有 } n \text { 列时，} T \text { 的定义域是 } \mathbb{R}^n$

$\displaystyle \text { 当 } A \text { 的每一列有 } m\text { 个 entries 时，}T \text { 的值域是 } \mathbb{R}^m$

判断向量是否属于线性变换的值域
Determine if a vector is in the range of a linear transformation.
几何描述向量在线性变换下的图像
Geometrically describe the image of a vector under a linear transformation.

剪切变换
放缩变换
旋转变换
镜像变换
投射变换
利用变换的线性性质求向量在变换下的图像
Use the linearity of transformations to find the images of vectors under the transformation.
判定关于线性变换陈述的有效性
Determine the validity of statements about linear transformations.

$\displaystyle A(\mathbf{u}+\mathbf{v})=A \mathbf{u}+A \mathbf{v} \text { and } A(c \mathbf{u})=c A \mathbf{u}$

$\displaystyle T(\mathbf{0})=\mathbf{0}$

$\displaystyle T(c \mathbf{u}+d \mathbf{v})=c T(\mathbf{u})+d T(\mathbf{v})$

1.9: 线性变换的矩阵
The Matrix of a Linear Transformation

求线性变换的标准矩阵

Find the standard matrix of a linear transformation.
求在线性变换下像为给定向量的原像向量
Find vectors whose images under a linear transformation are given.
判断关于线性变换性质的陈述是否正确
Determine the validity of statements about properties of linear transformations.
判断线性变换是否为单射或满射
Determine if linear transformations are one-to-one or onto.

^HW4^

1.10*: 商业、科学与工程中的线性模型
Linear Models in Business, Science, and Engineering

2.1: 矩阵运算
Matrix Operations

计算矩阵的和、积及标量积
Compute sums, products, and scalar products of matrices.
寻找满足给定矩阵乘积性质的矩阵值
Find values of matrices such that products of matrices have given properties.
判定关于矩阵运算的陈述是否成立
Determine the validity of statements about matrix operations.

2.2: 矩阵的逆
The Inverse of a Matrix

使用公式求 $2 \times 2$ 矩阵的逆矩阵
Find the inverse of a 2×2 matrix using the formula.
利用矩阵的逆求解线性方程组
Use the inverse of a matrix to solve a linear system.
判断关于矩阵逆的陈述是否正确
Determine the validity of statements about inverses of matrices.
求解涉及可逆矩阵的方程
Solve equations involving invertible matrices.
通过行约简法求矩阵的逆
Find the inverse of a matrix using row reduction.

2.3: 可逆矩阵的特征
Characterizations of Invertible Matrices

利用可逆矩阵定理判断关于矩阵的陈述是否成立
Use the Invertible Matrix Theorem to determine the validity of statements about matrices.
求线性变换的逆变换
Find the inverse of a linear transformation.

2.8*: Rn 的子空间
Subspaces of Rn

2.9*: 维度与秩
Dimension and Rank

^HW4^

3.1: 行列式简介
Introduction to Determinants

使用余子式展开计算行列式
Compute determinants using cofactor expansions.
确定初等行变换对行列式的影响
Determine the effect of elementary row operations on a determinant.
计算矩阵标量倍数的行列式
Calculate determinants of scalar multiples of matrices.

3.2: 行列式的性质
Properties of Determinants

识别行列式的性质
Identify properties of determinants
通过行化简为阶梯形求行列式
Find determinants by row reduction to echelon forms.
利用行列式的性质计算行列式
Use properties of determinants to evaluate determinants.
证明行列式的性质
Prove properties of determinants
用行列式判断矩阵是否可逆或向量组是否线性无关
Use determinants to determine if a matrix is invertible or a set of vectors is linearly independent
判断关于行列式性质的命题是否正确
Determine the validity of statements about properties of determinants.

^HW5^

4.1: 向量空间与子空间
Vector Spaces and Subspaces

判断给定集合是否为向量空间
Determine whether a given set is a vector space.
判断给定集合是否为子空间
Determine whether a given set is a subspace.
判断关于向量空间和子空间陈述的有效性
Determine the validity of statements about vector spaces and subspaces.
解决涉及向量空间和子空间的应用问题
Solve applications involving vector spaces and subspaces.

4.2: 零空间、列空间与线性变换
Null Spaces, Column Spaces, and Linear Transformations

判断一个向量是否位于矩阵的零空间或列空间中
Determine whether a vector is in the null or column space of a matrix.
在矩阵的零空间、列空间或行空间中找到一个非零向量
Find a nonzero vector in the null, column, or row space of a matrix.
判断关于线性变换及零空间、列空间和行空间的陈述是否正确
Determine the validity of statements about linear transformations and null, column, and row spaces.
利用零空间和列空间的理论来求解线性系统的解
Use the theory of null and column spaces to find solutions of linear systems.

4.3: 线性无关集；基底
Linearly Independent Sets; Bases

判断一个集合是否线性无关，是否为向量空间的基，或是否张成向量空间
Determine whether a set is linearly independent, or is a basis for, or spans a vector space.
寻找矩阵的零空间、列空间或行空间的基
Find bases for the null, column, or row space of a matrix.
寻找一组向量张成空间的基
Find a basis for the span of a set of vectors.
判断关于线性无关集和基的陈述的正确性
Determine the validity of statements about linearly independent sets and bases.

4.5: 向量空间的维度
The Dimension of a Vector Space

寻找矩阵子空间的维度
Find the dimension of a subspace.
寻找矩阵的零空间、列空间和行空间的维度
Find the dimensions of the null, column, and row spaces for a matrix.

4.6*: 基变换
Change of Basis

^HW6^

– – -MID 2- – –

5.1: 特征向量和特征值
Eigenvectors and Eigenvalues

判断一个向量或数是否为给定矩阵的特征向量或特征值
Determine if a vector or number is an eigenvector or eigenvalue of a given matrix.
找到对应于某个特征值的特征空间的基
Find a basis for the eigenspace corresponding to an eigenvalue.
求矩阵的特征值
Find the eigenvalues of matrices.
判断关于特征向量和特征值的陈述是否正确
Determine the validity of statements about eigenvectors and eigenvalues.

5.2: 特征方程
The Characteristic Equation

求一个 $2 \times 2$ 矩阵的特征多项式及特征值
Find the characteristic polynomial and eigenvalues of a 2×2 matrix.
求一个 $3 \times 3$ 矩阵的特征多项式
Find the characteristic polynomial of a 3×3 matrix.
求三角矩阵的特征值
Find the eigenvalues of triangular matrices.

^HW7^

5.3: 对角化
Diagonalization

计算 $A^k$ 对于 $A=P D P^{-1}<br>$ 的值
使用对角化定理求矩阵的特征值及每个特征空间的基
对矩阵进行对角化
判断关于对角化的陈述是否有效
判定矩阵是否可对角化
回答关于矩阵对角化的概念性问题

5.4*: 特征向量与线性变换
Eigenvectors and Linear Transformations

6.1: 内积、长度与正交性
Inner Product, Length, and Orthogonality

使用内积对向量进行运算
Perform operations on vectors using inner products.
找到一个单位向量
Find a unit vector
找出两个向量之间的距离
Find the distance between two vectors
判断两个向量是否正交
Determine whether two vectors are orthogonal

6.2: 正交集
Orthogonal Sets

判断一组向量是否正交
Determine whether a set of vectors is orthogonal.
找到通过给定向量和原点的直线的正交投影
Find an orthogonal projection onto a line through a given vector and the origin.
将一个向量表示为两个正交向量的和
Write a vector as a sum of two orthogonal vectors.
求向量与通过原点的直线之间的距离
Find the distance between a vector and a line through the origin.
判断一组向量是否标准正交
Determine whether a set of vectors is orthonormal.
判断关于正交集合或矩阵及正交投影的陈述的有效性
Determine the validity of statements about orthogonal sets or matrices and orthogonal projections.

6.3: 正交投影
Orthogonal Projections

求向量的正交分解
Find the orthogonal decomposition of a vector.
求给定子空间上的正交投影
Find an orthogonal projection onto a given subspace.
使用最佳逼近定理求最近点或距离
Use the best approximation theorem to find the closest point or a distance.
构造与给定正交集正交的向量
Construct a vector that is orthogonal to a given orthogonal set.
判断关于子空间正交投影陈述的正确性
Determine the validity of statements about orthogonal projections onto subspaces.

^HW8^

6.4: 格拉姆-施密特过程
The Gram-Schmidt Process

使用格拉姆-施密特过程生成正交基，并找到正交基
Use the Gram-Schmidt process to produce an orthogonal basis and find an orthonormal basis.

^HW9^

6.5: 最小二乘问题
Least-Squares Problems

使用正规方程求系统的最小二乘解
Use normal equations to find least-squares solutions of systems.
判断关于最小二乘解陈述的有效性
Determine the validity of statements about least-squares solutions.

6.6*: 线性模型的应用
Applications to Linear Models

^HW10^

7.1: 对称矩阵的对角化
Diagonalization of Symmetric Matrices

判定关于对称矩阵陈述的有效性
Determine the validity of statements about symmetric matrices.
运用对称矩阵的谱定理寻找对称矩阵的谱分解
Use the Spectral Theorem for Symmetric Matrices to find a spectral decomposition of symmetric matrices.

7.4: 奇异值分解
The Singular Value Decomposition

求矩阵的奇异值
Find the singular values of a matrix.
求矩阵的奇异值分解
Find a singular value decomposition of a matrix.
利用矩阵的奇异值分解求特定值或向量
Use the singular value decomposition of a matrix to find particular values or vectors.

– – -FINAL- – –
$\left[\begin{array}{l} x_1(t) \\ x_2(t) \end{array}\right]=\left[\begin{array}{l} c_1 e^{3 t} \\ c_2 e^{-5 t} \end{array}\right]=c_1\left[\begin{array}{l} 1 \\ 0 \end{array}\right] e^{3 t}+c_2\left[\begin{array}{l} 0 \\ 1 \end{array}\right] e^{-5 t}$

Cementine Cafe

220大纲中文版

1.1: 线性方程组
Systems of Linear Equations

1.2: 行简化与阶梯形态
Row Reduction and Echelon Forms

1.3: 向量方程
Vector Equations

1.4: 矩阵方程 Ax=b
The Matrix Equation Ax = b

1.5: 线性系统的解集
Solution Sets of Linear Systems

1.6*: 线性系统的应用
Applications of Linear Systems

1.7: 线性无关
Linear Independence

– – -MID 1- – –

1.8: 线性变换简介
Introduction to Linear Transformations

1.9: 线性变换的矩阵
The Matrix of a Linear Transformation

1.10*: 商业、科学与工程中的线性模型
Linear Models in Business, Science, and Engineering

2.1: 矩阵运算
Matrix Operations

2.2: 矩阵的逆
The Inverse of a Matrix

2.3: 可逆矩阵的特征
Characterizations of Invertible Matrices

2.8*: Rn 的子空间
Subspaces of Rn

2.9*: 维度与秩
Dimension and Rank

3.1: 行列式简介
Introduction to Determinants

3.2: 行列式的性质
Properties of Determinants

4.1: 向量空间与子空间
Vector Spaces and Subspaces

4.2: 零空间、列空间与线性变换
Null Spaces, Column Spaces, and Linear Transformations

4.3: 线性无关集；基底
Linearly Independent Sets; Bases

4.5: 向量空间的维度
The Dimension of a Vector Space

4.6*: 基变换
Change of Basis

– – -MID 2- – –

5.1: 特征向量和特征值
Eigenvectors and Eigenvalues

5.2: 特征方程
The Characteristic Equation

5.3: 对角化
Diagonalization

5.4*: 特征向量与线性变换
Eigenvectors and Linear Transformations

6.1: 内积、长度与正交性
Inner Product, Length, and Orthogonality

6.2: 正交集
Orthogonal Sets

6.3: 正交投影
Orthogonal Projections

6.4: 格拉姆-施密特过程
The Gram-Schmidt Process

6.5: 最小二乘问题
Least-Squares Problems

6.6*: 线性模型的应用
Applications to Linear Models

7.1: 对称矩阵的对角化
Diagonalization of Symmetric Matrices

7.4: 奇异值分解
The Singular Value Decomposition

– – -FINAL- – –
$\left[\begin{array}{l} x_1(t) \\ x_2(t) \end{array}\right]=\left[\begin{array}{l} c_1 e^{3 t} \\ c_2 e^{-5 t} \end{array}\right]=c_1\left[\begin{array}{l} 1 \\ 0 \end{array}\right] e^{3 t}+c_2\left[\begin{array}{l} 0 \\ 1 \end{array}\right] e^{-5 t}$

220大纲 中文版

1.1: 线性方程组Systems of Linear Equations

1.2: 行简化与阶梯形态Row Reduction and Echelon Forms

1.3: 向量方程Vector Equations

1.4: 矩阵方程 Ax=bThe Matrix Equation Ax = b

1.5: 线性系统的解集Solution Sets of Linear Systems

1.6*: 线性系统的应用Applications of Linear Systems

1.7: 线性无关Linear Independence

– – -MID 1- – –

1.8: 线性变换简介Introduction to Linear Transformations

1.9: 线性变换的矩阵The Matrix of a Linear Transformation

1.10*: 商业、科学与工程中的线性模型Linear Models in Business, Science, and Engineering

2.1: 矩阵运算Matrix Operations

2.2: 矩阵的逆The Inverse of a Matrix

2.3: 可逆矩阵的特征Characterizations of Invertible Matrices

2.8*: Rn 的子空间Subspaces of Rn

2.9*: 维度与秩Dimension and Rank

3.1: 行列式简介Introduction to Determinants

3.2: 行列式的性质Properties of Determinants

4.1: 向量空间与子空间Vector Spaces and Subspaces

4.2: 零空间、列空间与线性变换Null Spaces, Column Spaces, and Linear Transformations

4.3: 线性无关集；基底Linearly Independent Sets; Bases

4.5: 向量空间的维度The Dimension of a Vector Space

4.6*: 基变换Change of Basis

– – -MID 2- – –

5.1: 特征向量和特征值Eigenvectors and Eigenvalues

5.2: 特征方程The Characteristic Equation

5.3: 对角化Diagonalization

5.4*: 特征向量与线性变换Eigenvectors and Linear Transformations

6.1: 内积、长度与正交性Inner Product, Length, and Orthogonality

6.2: 正交集Orthogonal Sets

6.3: 正交投影Orthogonal Projections

6.4: 格拉姆-施密特过程The Gram-Schmidt Process

6.5: 最小二乘问题Least-Squares Problems

6.6*: 线性模型的应用Applications to Linear Models

7.1: 对称矩阵的对角化Diagonalization of Symmetric Matrices

7.4: 奇异值分解The Singular Value Decomposition

– – -FINAL- – –

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220大纲中文版

1.1: 线性方程组
Systems of Linear Equations

1.2: 行简化与阶梯形态
Row Reduction and Echelon Forms

1.3: 向量方程
Vector Equations

1.4: 矩阵方程 Ax=b
The Matrix Equation Ax = b

1.5: 线性系统的解集
Solution Sets of Linear Systems

1.6*: 线性系统的应用
Applications of Linear Systems

1.7: 线性无关
Linear Independence

1.8: 线性变换简介
Introduction to Linear Transformations

1.9: 线性变换的矩阵
The Matrix of a Linear Transformation

1.10*: 商业、科学与工程中的线性模型
Linear Models in Business, Science, and Engineering

2.1: 矩阵运算
Matrix Operations

2.2: 矩阵的逆
The Inverse of a Matrix

2.3: 可逆矩阵的特征
Characterizations of Invertible Matrices

2.8*: Rn 的子空间
Subspaces of Rn

2.9*: 维度与秩
Dimension and Rank

3.1: 行列式简介
Introduction to Determinants

3.2: 行列式的性质
Properties of Determinants

4.1: 向量空间与子空间
Vector Spaces and Subspaces

4.2: 零空间、列空间与线性变换
Null Spaces, Column Spaces, and Linear Transformations

4.3: 线性无关集；基底
Linearly Independent Sets; Bases

4.5: 向量空间的维度
The Dimension of a Vector Space

4.6*: 基变换
Change of Basis

5.1: 特征向量和特征值
Eigenvectors and Eigenvalues

5.2: 特征方程
The Characteristic Equation

5.3: 对角化
Diagonalization

5.4*: 特征向量与线性变换
Eigenvectors and Linear Transformations

6.1: 内积、长度与正交性
Inner Product, Length, and Orthogonality

6.2: 正交集
Orthogonal Sets

6.3: 正交投影
Orthogonal Projections

6.4: 格拉姆-施密特过程
The Gram-Schmidt Process

6.5: 最小二乘问题
Least-Squares Problems

6.6*: 线性模型的应用
Applications to Linear Models

7.1: 对称矩阵的对角化
Diagonalization of Symmetric Matrices

7.4: 奇异值分解
The Singular Value Decomposition

– – -FINAL- – –
$\left[\begin{array}{l} x_1(t) \\ x_2(t) \end{array}\right]=\left[\begin{array}{l} c_1 e^{3 t} \\ c_2 e^{-5 t} \end{array}\right]=c_1\left[\begin{array}{l} 1 \\ 0 \end{array}\right] e^{3 t}+c_2\left[\begin{array}{l} 0 \\ 1 \end{array}\right] e^{-5 t}$