24.01.001 Vector Equations

1 Introduction to Vectors

1.1 向量

定义 1.1.1 向量

中学阶段只讨论向量的几何意义，由此我们只能想象出二维和三维的向量。从代数角度，我们可以定义 \(n\) 维的向量。

定义笛卡尔积为两个集合的运算 \(A\times B=\{(u,v)\mid u\in A,v\in B\}\)，其元素是二元组 \((u,v)\)。同理若元素 \(u\) 本身即为一个二元组，那么 \((u,v)\) 同样可以写成三元组 \((u_1,u_2,v)\)。由此可以递归定义 \(n\) 元组 \((a_1,a_2,\dots ,a_n)\)，我们称这个 \(n\) 元组为一个 \(n\) 维向量，记作 \(\mathbf{a}=(a_1,a_2\dots a_n)\)。这里用黑体区分数和向量，但以后的笔记中为了方便可以会略去。通常会把向量写成方括号括起来，用一个 \(n\times 1\) 矩阵来描述的形式，\(\mathbf a=\begin{bmatrix}a_1\\a_2\\\dots\\a_n\end{bmatrix}\)。

定义 1.1.2 加法和数乘 向量的加法定义为每个元素对应的加法，数乘为每个元素乘法。

\[\mathbf a+\mathbf b=\begin{bmatrix}a_1+b_1\\a_2+b_2\\\dots\\a_n+b_n\end{bmatrix}\\c\mathbf a=\begin{bmatrix}ca_1\\ca_2\\\dots\\ca_n\end{bmatrix} \]

多数情况下，向量满足加法交换律和数乘分配律，即这是一个交换环。

定义 1.1.3 线性组合

这里先粗浅地把向量数乘与加法的组合，称作向量的线性组合。注意零向量的存在，因为，\(0\mathbf v=\mathbf 0\)。

例如，\(w=cv+du\) 就是 \(u,v\) 的一个线性组合。

从几何角度来看，对于两个二维向量 \(u,v\)，如果 \(u\ne kv\)，显然 \(u,v\) 的线性组合可以组合出所有平面上的点（用点代表向量）。否则，只能表示一条线（这肯定是一条过原点的线）。这在三维、四维甚至更高维度是依然是成立的。同理可知，用 \(n\) 个向量，如果其中任意一个向量无法被其余向量线性组合出来，那么这 \(n\) 个向量可以组合出 \(n\) 维空间，即 \(\mathbb R^3\)。*

difinition span

If \(v_1,\dots,v_p\) are in \(\mathbb R^n\), then set of all linear combinations of \(v_1,...,v_p\) is denoted by \(\mathrm{Span}\{v_1,\dots,v_p\}\) and is called the subset of \(\mathbb R^n\) spanned by \(v_1,\dots,v_p\).

习题

1.27 求 \(n\) 维洞见中的 \(m\) 维超立方体个数。

根据组合意义，得 \(\dfrac{2^n\binom{n}{m}}{2^m}\).

1.2 模和点乘

定义 1.2.1 点乘

对于两个同为 \(\mathbb R^n\) 中的向量 \(v,u\)，记其点乘为：

\[v\cdot u=\sum_{i=1}^{n}v_i\times u_i \]

假如两个向量点积为 \(0\)，则称这两个向量垂直（perpendicular）。在二维平面上也就是对应 \(90\) 度角。

向量点乘满足交换律，由实数乘法具有交换律显然得到。

定义 1.2.2 模

记 \(v\cdot v\) 再开根号为向量 \(v\) 的模，记作 \(||v||^2=v\cdot v=\sum_{i=1}^{n}v_i^2\)。

定义 1.2.3 单位向量

定义模长为 \(1\) 的向量为单位向量。例如，二维平面上，向量 \(u=(\cos \theta,\sin \theta)\) 就是一个单位向量。

定理 1.2.1 将任意一个向量除以它的模，将会得到单位向量。

定理 1.2.2 勾股定理

若 \(v\cdot w=0\)，则 \(||v||^2+||w||^2=||v-w||^2\)。

证明 展开得到 \(v_1^2+v_2^2+w_1^2+w_2^2=(v_1-w_1)^2+(v_2-w_2)^2\)，这是显然的。

1.3 矩阵

矩阵运算

把若干列向量拼在一起就形成了矩阵。

定义 1.3.1 矩阵加法

矩阵的加法即为每个位置对应相加，或者可以看成向量的加法。

定义 1.3.2 矩阵乘法

记 \(A\times B=C\)，其中 \(A\) 是一个 \(m\times n\) 的矩阵，\(B\) 是一个 \(n\times p\) 的矩阵。有 \(C\) 是一个 \(m\times p\) 的矩阵。

1. 常规方法

\(C\) 中每个位置的数 \(c_{ij}\) 可以写成 \(\sum_k a_{ik}b_{kj}\)。

2. \(A\) 矩阵列向量的线性组合

3. \(B\) 矩阵行向量的线性组合

4. 用列和行的乘积表示

\(AB=\sum\limits_{i=1}^{n} col_i(A)\times row_i(B)\)。

不同的对矩阵乘法方式的理解可以在不同的时候派上用场。例如，向量的线性组成就可以写成矩阵乘以向量的形式。

差分矩阵，\(A\)，满足 \(Ax=\begin{bmatrix}x_1\\x_2-x_1\\x_3-x_2\end{bmatrix}\)。

构造 \(A=\begin{bmatrix}1 & 0 & 0\\-1 & 1 & 0\\ 0 & -1 & 1\end{bmatrix}\).

2 Linear Equations

2.1 SYSTEMS OF LINEAR EQUATIONS

definition linear equations : about some variables \(x_1,x_2,\dots ,x_n\), and coefficients \(a_1,a_2,\dots ,a_n\) which are real or complex numbers known in advance. a equation written like :

\[a_1x_2+a_2x_2+\dots+a_nx_n=b \]

\(b\) is called constant.

A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables.

The matrix formed by coefficients is called cofficent matrix.

The set of all possible solutions is called the solution set of the linear system. \(\emptyset\), \(1\) or \(\infty\).

definition Augmented Matrix

An augmented matrix of a system consists of the coefficient matrix with an added column containing the constants from the right sides of the equations.

2.2 Solve a System of Linear Equations

ELEMENTARY ROW OPERATIONS

1. (Replacement) Replace one row by the sum of itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.

Obviously, ROW/COLUMN operations don't change the solution of the equations and all these ioerations is invertible.

This methos is called The Gauss's Method.

definition Echelon Matrix(Reduced Echelon Matrix)

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1. All nonzero rows are above any rows of all zeros.
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
3. All entries in a column below a leading entry are zeros.

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

4. The leading entry in each nonzero row is 1.
5. Each leading 1 is the only nonzero entry in its column.

theorem Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon matrix.

The reduced row echelon form is unique for every matrices. So if \(2\) matrices have the same reduced row echelon form, we call them equivalent. Noting as \(A\sim B\).

definiton Pivot Position

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is a column of A that contains a pivot position.

theorem Existence and Uniqueness Theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column.

variables correspnding to pivot columns are called basic variables. Others are called free variables.

Existence and Uniqueness Theorem

A linear system is consistent(相容) if and only if the rightmost column of the augmented matrix is not a pivot column. that is, if and only if an echelon form of the augmented matrix has no row of the for \(\begin{bmatrix}0 & \dots & 0 & b\end{bmatrix}\) with \(b\) nonzero. If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

2.3 SOLUTION SETS OF LINEAR SYSTEMS

definition Homogeneous Linear systems

A system of linear equations is said to be homogeneous if it can be written in the form \(Ax = 0\).

Of course there is always at least one solution \(x=\mathbf{0}\)(the zero vector in \(\mathbb R ^n\)). Which is usually called the trivial solution.

theorem The homogeneous equation \(Ax=0\) has a nontrivial solution if and only if the equation has at least one free variable.

Consider \(v_h\) is a solution of equation \(Ax=0\), and \(p\) is a solution of equation \(Ax=b\). Hence all solutions of this function is in the form \(w=p+v_h\).

WRITING A SOLUTION SET IN PARAMETRIC VECTOR FORM

Also called parametric descriptions.

Row reduce the augmented matrix to reduced echelon form.
Express each basic variable in terms of any free variables appearing in an equation.
Write a typical solution x as a vector whose entries depend on the free variables, if any.
Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters.

2.4 LINEAR INDEPENDENCE

An indexed set of vectorsfv1;:::;vpginRn is said to belinearly independent if the vector equation \(x_1v_1+x_2v_2+\dots x_pv_p=0\) has only the trivial solution.The set \(v_1,\dots,v_p\) is said to belinearly dependent if there exist weights \(c_1,\dots,c_p\), not all zero, such that \(c_1v_1+c_2v_2+\dots c_pv_p=0\).

An indexed set \(S=\{v_1,v_2,\dots,v_p\}\) of two or more vectors is linearly dependent if and only if at least one of the vectors in \(S\) is a linear combination of the others. In fact, if \(S\) is linearly dependent and \(v_1\ne 0\), then some \(v_j\) (with \(j>1\)) is a linear combination of the preceding vectors, \(v_1,v_2,\dots v_j\).

2.5 LINEAR TRANSFORMATION

A transformation(or function or mapping)\(T\) from \(R^n\) to \(R^m\) is a rule that assigns to each vector \(x\) in \(R^n\) a vector \(T\). \(x\in R^m\).The set \(R^n\) is called the domain of \(T\), and \(R^m\) is called the codomain of \(T\). The notation \(T: R^n\to R^m\) indicates that the domain of T is \(R^n\) and the codomain is \(R^m\).For \(x\in R^n\),the vector \(Tx\in R^m\) is called the image of \(x\) (under the action of \(T\)).The set of all images \(Tx\) is called the range of \(T\).

A transformation (or mapping)\(T\) is linear if:
(i) \(T(u+v)=T(u)+T(v)\) for \(all u,v\) in the domain of \(T\);
(ii) \(T(cu)=cT(u)\) for all scalarscand all \(u\) in the domain of \(T\).

How to compute the matrix of a linear transformation? Actually, assume \(T\) is a linear transformation, we can find \(A\) using the identity matrix \(I_n\).

\[A=T(I_n)=\begin{matrix}T(e_1) & T(e_2) & \dots & T(e_n)\end{matrix} \]

A mapping \(T: R^n\to R^m\) is said to be onto \(R^m\) if each \(b\) in \(R^m\) is the image of at least one \(x\) in \(R^n\).

A mapping \(T:R^n\to R^m\) is said to be one-to-one if each \(b\) in \(R^m\) is the image of at most one \(x\) in \(R^n\).

Theorem 11 \(T\) is one-to-one if and only if \(Ax=0\) only has the trival solution(Also say the column vectore of \(A\) if linear independent).

Theorem 12 \(T\) maps \(\mathbb R^n\) onto \(\mathbb R^m\) if and only if the columns of \(A\) span \(\mathbb R^m\);