MIT18.06_2_Unit 1 AX=B and the four subspaces

Note:

the properties of A and b determine the solutions x (if any exist) and pay particular attention to the solutions to Ax=0 ;

the geometry of linear equations

there are THREE ways of thinking about these systems.

• row method: individual equations
• column method: combining the columns
• matrix method: more compact and powerful way of describing systems of linear equations.

Ax = b
n linear equations, n unknowns.
n equal 2, for example:

row picture

被认为是n个(n-1)维度空间的交集

比如说，二维矩阵方程（二维空间为平面）的解是俩一维空间（直线）的交集。
三维矩阵方程（rank3，其实这里说是 秩 更合适）的解，是三个二维空间（平面）的交集

colomn picture

可以看作是 n 个 n 维向量与未知量的线性组合

Matrix Multiplication

How do we multiply a matrix A by a vector x

there are 2 methods to get the entries of x as the coefficients of a linear combination of the column vectors of the matrix:

this technique shows that Ax is a linear combination of the columns of A.

you may also calculate the product Ax by taking the dot product of each row of A with the vector x:

Linear Independence

for:

• for every possible vector?
• do the linear combinations of the column vectors fill the xy-plane (or space, in the three dimensional case)?

NO, if A is a singular matrix. In this singualr case its column vectors are linearly dependent.

All linear combinations of those vectors lie on a point or line(in 2 dimensions) or on a point, line or plane(in three dimensions). The combinations don’t fill the whole space.

Overview of key ideas

The pro. Strang give us 2 examples to understand a series of conceptions.

The first example:

the other example:

主要用这两个例子，展示了 基、子空间、长成空间 的概念

Subspaces

• 从几何上看，矩阵C的各列所表征的向量都在一个平面上（这些向量组是线性相关的，来自矩阵A点向量组线性不相关）

• 三维向量空间中包含许多向量，可以用向量组A表示，但是无法用C表示，这些无法用矩阵C中的向量组表示的点

  • 不在C所在平面上
  • 无法用 向量组C 的线性组合表示

basis

• a basis foris a collection ofindependent vectors in.
• a basis is a collection ofvectors whose combinations cover the whole space.
• a collection of vectors forms a basis whenever a matrix which has those vectors as its columns is invertible.

vector space

a vector space is a collection of vectors that is closed under linear combinations.

向量空间 就是 线性组合闭合的向量集合

subspaces

a subspace is a vector space inside another vector space

子空间是 被一个向量空间内含的向量空间.

The subspaces ofare:

• the origin.
• a line through the origin.
• a plane through the origin.
• all of.

Elimination with matrices

is the technique most commonly used by computer software to solve systems of linear equations.

Method of Elimination

消去法(Elimination) 可找到的解，在矩阵可逆时。

这里有个概念叫做（主元），指着每个行列数最小的非零值。随着矩阵的不断消除，若矩阵出现某行不存在 pivot，那么原矩阵不可逆。

Back Substitution

当通过消去法或者上三角举证后，可以利用(倒转代换)的方法得到方程的解。

得到矩阵后可以通过方程，以的顺序迭代，得到最后的结果。

Elimination Matrices

消去矩阵，是一种基本矩阵，若要消除 row 2 column 1 的元素（通过行变换）可以左乘矩阵。相应的，通过列变换，右乘举证。

Permutation Matrices

置换矩阵，是另一种基本矩阵，左乘则于行变换作比较，右乘以列变换做比较。
For example,

基本矩阵共同的重要性质： 可逆
具体就是做基本变换的反变换。

Multiplication and inverse matrice

我们从四个角度去思考，两个矩阵的乘积。

Standard

standrad way:equals the dot product of row i of matrix A and column j of the matrix B.

这就是大家在线性代数中学过的，最终矩阵的元素定义。

Columns

the product of matrix A and column j of matrix B equals column j of matrix C. This tells us that the columns of C are combinations of columns of A.

矩阵和 矩阵的第列 的乘积（其实就是我们之前提及的，），可以视为 向量组的线性组合。

Rows

The product of rowof matrix A and matrix B equals rowof matrix C. So the rows of C are combinations of rows of B.

矩阵的第列和矩阵的乘积为矩阵, 这时的行为行的线性组合。

Column times row

A column ofis anvector.
A row ofis avector.
Their product is a matrix:

The columns of this matrix are multiples of the column ofand the rows are multiples of the row of.
The product ofandis the sum of these “column times row” matrices:

可以类比standrad视角的书写方法，每个元素的被一并提出。

Block

If we subdivideandinto blocks that match properly, we can write the productin terms of products of the blocks:

Here

老师说，即使是高斯，也不能一眼看出这里的问题所在。

Inverse(Square matrices)

Ifis a square matrix, the most important question you can ask about it has an inverse. If it does, thenand we say thatis invertible or nonsigular.

Ifis singular -i.e.does not have an inverse - its determinat is zero and we can find some non-zero vectorfor which.

• 方阵如果存在 方阵使得，那么称这个方阵为矩阵的逆。
  • 如果矩阵可逆，则称原矩阵为可逆或者是非奇异。
  • 如果矩阵不可逆，则称原矩阵为不可逆或者是奇异。
• 如果矩阵奇异-也就是说没有逆阵-这个矩阵的行列式不为零。

Finding the inverse of a matrix is closely related to solving systems of linear equations.

Gauss-Jordan Elimination

我们可以使用

• Gauss-Jordan Elimination(有一种翻译叫作高斯-若而当消元法)
  • 可以同时解决 两个及以上 的线性方程。

Just augment the matrix with the whole identity matrix:

矩阵方程，可以很自然的表示多个线性方程组，并且通过 Gauss-Jordan Elimination 完成消除，这也就完成了 同时解多个方程组。

使用 单位矩阵 进行矩阵增广，得到 增广矩阵。
** augment 在某些语境下，有 强化\增强 的意思 **

(Once we have used Gauss’ elimination method to convert the original matrix to upper triangular form, we go on to use Jordan’s idea of eliminating entries in the upper right portion of the matrix.)

当我们完成使用 Gauss消元法 将上三角矩阵形式，紧接着运用 Jordan的想法 消除矩阵右上三角位置处的元素。

Guass-Jordan 证明

we can write the results of the elimination method as the product of a number of elimination matriceswith the matrix.
Lettingbe the product of all the, we write the result of this Gauss-Joran elimination using block matrices:

But if, then.

我们把消去法的效果等效于左乘消去矩阵, 对增光矩阵进行块划分，等效于对两个子矩阵和一起左乘

其中：，
根据，矩阵的逆的定义，可以直接得出

Factoration A = LU

• understand Gaussian elimination in terms of matrices.
• find a matrixsuch that.

Inverse&Transpose of a product

The inverse of a matrix productis.

transpose of a matrix by exchanging its rows and columns.
the entry in row i column j of A is the entry in row j column i of

The transpose of a matrix productis. For any invertible matrix, the inverse ofis

A = LU

We’ve seem how to use elimination to convert a suitable matrixinto an upper triangular matrix.

我们将会见证如何将一个合适的矩阵转换为一个上三角矩阵。

when there are no row exchanges, we can describe the elimination of the entries of matrixin terms of multiplicaiton by a succession of elimination matrices, so that.

In the two by two case this looks like:

从左到右，依次是矩阵

we can convert this to a factorizationby “canceling” the matrix;
multiply by its inverse to get

从左到右，依次是矩阵

The matrixis upper trangular with pivots on the diagonal.
矩阵是一个上三角矩阵（因为Gaussian Elimination只对主对角线以下的下三角区域元素进行消元）

进一步可以进行列变换，生成下三角矩阵x对焦矩阵x上三角矩阵 的形式:

从左到右，依次为矩阵

how expensive is elimination

Some applicaitons require inverting very large matrices.

许多应用，需要对超大矩阵求逆

When using elimination to find the factorizationwe just saw that we can buildas we go by keeping track of row subtractions. We have to rememberand (the matrix which will become);

当使用 消去法 寻找分解时，可以知道，利用消去法的track来循迹。

we don’t have to storeorin the computer’s memory.

我们不得不，存储矩阵和矩阵于计算机内存。

How many operations does the computer perform during the elimination process for anmatrix ?

A typical operation is to multiply one row and then subtract it from another, which requires on the order ofoperations. There arerows, so the total number of operations used in eliminating entries in the first column is about. The second row and column are shorter; that product costs aboutoperations, and so on. The total number of operations needed to factorintois on the order of:

While we’re factoringwe’re also operating on b. That costs aboutoperations, which is hardly worth counting compared to

Row exchanges

这一节就不翻译了，大概就是，行交换矩阵的表达形式。

What if there are row exchanges? In other words, what happens if there’s a zero in a pivot position?
To swap two rows, we multiply on the left by a permutation matrix. For example,

swaps the first and second rows of amatrix. The inverse of any permutation matrixis.

There aredifferent ways to permute the rows of anmatrix (including the permutation that leaves all rows fixed) so there arepermutation matrices. These matrices form a multiplicative group.

Homogeneous Linear System

这一节是TA的课程，推荐听听 https://www.youtube.com/watch?v=3cMyj8EKFGo&t=11s （需要科学上网
比较直观的解释了，非齐次/齐次 方程的任意解的形式。

The setof points

• 集合满足方程。
  • 集合中的所有点都满足方程
  • 方程的所有解都在集合之中。
• 约束方程是 非齐次线性方程
  • 方程（组）的左侧，为线性的。
  • 方程（组）的右侧（常数项）不为零。

The setof points

• 约束方程是 齐次线性方程
  • 方程（组）的左侧，为线性的。
  • 方程（组）的右侧（常数项）为零。

集合对应的几何形态为平面，此外平面方程和平行。
另外，齐次线性方程经过原点：即，平面经过原点坐标原点。

非齐次线性方程解的一般形式

Martina(TA) 把解的形式写成了上状

理解：

• 等式右边第一个矩阵，是 平面上的一个特解
• 等式右边的后两个矩阵
  • 线性无关
• 的解释
  • 两个线性无关的向量的 线性组合 可以表示平面中的任意点
• 再加上 原平面 上的一个特解，等效于将原点平移到的特解位置

助教的图像解释：

Transposes and Permutations

Transposes

Permutations: execute row exchanges.

Multiplication by a permutation matrixswaps the rows of a matrix; when applying the method of elimination we use permutation matrices to move zeros out of pivot positions. Our factorizationthen becomes, whereis a permutation matrix which reorders any number of rows of. Recall that, i.e. that.

• 乘上转换 矩阵可以交换对应的 矩阵行；
• 在 消去法 的应用中，可以利用矩阵 进行主元的变化；

Permutation

When we take the transpose of a matrix, its rows become columns and its columns become rows. If we denote the entry in row i column j of matrixby, then we can describeby:, then we can describeby:. For example:

symmetric matrix:

A matrixis symmetric if. Given any matrix(not necessarily square) the productis always symmetric, because.(Note that)

Matlab not only checks whether that pivot is not zero. It checks for is that pivot big enough, because it doesn’t like very, very small pivots. Pivots close to zero are numerically bad.

• Matlab 不仅检查主元是否为零。
• 它还确认主元是否 够大。
  • 如果 主元 过小，从 数值计算 角度，计算误差会比较大。
  • 从 代数 角度看，较小的 主元 其实没啥事。

对于阶行列式，仅仅因为 Permutation, 有种不同的变换矩阵。

超级重要的一个定理: