MIT18.06_3_A Bigger Picture with vector spaces

Not just vectors, but spaces of vectors and sub-spaces of those spaces.

不只是向量，还包括向量空间和这些空间的子空间。

Vector spaces

Let we say again what this word space is meaning.
When we say that word space, that means that I’ve got a bunch of vectors, a space of vectors. But not just any bunch of vectors.
It has to be a space of vectors – has to allow me to do the operations that vectors are for. It have to be able to add vectors and multiply by numbers.

谈及 空间 时，意味着 一大堆的向量。
但不是 一大堆向量 就能被称为 向量空间 ，必须满足，空间中的向量，能进行线性操作。

上面的内容是根据Gilbert老师的原话翻译的，有些不准确，准确翻译:

线性空间 应该满足，向量的线性组合，闭合，即向量的 线性组合 依然满足空间的条件。

One such vector space is, the set of all vectors with exactly two real number components. We depict the vectorby drawing an arrow from the origin to the pointwhich isunits to the right of the origin andunits above it, and we callthe “x-y plane”.

Another example of a space is, the set of (column) vectors withreal number components.

Closure

闭合性

The collection of vectors with exactly two positive real valued components is not a vector space.
The sum of any two vectors in that collection is again in the collection, but multiplying any vector by, say, -5, gives a vector that’s not in the collection.
We say that this collection of positive vectors is closed under addition but not under multiplication.

如果全部由 正实向量 组成空间，经过 加性运算 的结果也属于这个空间，但是考虑 数乘（multiplication） 不一定属于这个空间。
那么对于这个 space 来说，只 加法闭合 但没有 乘法闭合。

If a collection of vectors is closed under linear cominations (i.e. under addtion and multiplication by any real numbers), then we call that collection a vector space.

Subspaces

子空间

A vector space that is contained inside of another vector space is called a subspace of that space.
For example, take any non-zero vectorin. Then the set of all vectors, whereis a real number, forms a subspace of. This collection of vectors describes a line throughinand is closed under addition.

• 被一个向量空间包含的向量空间，叫作的 子空间。
• The subspaces ofare,

1. all of,
2. any line throughand
3. the zero vector alone(Z).

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• The subspaces ofare:

1. all of,
2. any plane through the origin,
3. any line through the origin, and
4. the zero vector alone(Z).

Column space

Given a matrixwith columns in, these columns and all their linear combinations form a subspace of. This is the column space.

If, the column space ofis the plane through the origin incontainingand.

Column space and nullspace

It’s really getting to the center of linear algebra.

Review of subspaces

• A vector space is a collection of vectors which is closed under linear combinations.
• for any 2 vectors v and w in the space and any 2 real numbersand, the vectoris also in the vector space.
• A subspace is a vector space contained inside a vector space.

存在两个过 原点 的子空间和，和的并集不一定是子空间，和的交集 一定是 子空间。

第一个证明 举反例。
对于与的交集，同时处于两个子空间中。那么他的线性组合，也 即在子空间中，也在子空间中。 那么也满足 线性运算闭合 特性。

Column space of A

The column space of a matrixis the vector space made up of all linear combinations of the columns of.

Solving

Given a matrix, for what vectorsdoeshave a solution?

QUESTION: 给定矩阵，对于任意向量，都有解?

Let

• 方程对于任意向量不一定有解。
• 解决方程等效于求解一个有 3个未知数的4等式方程
  • 如果存在 解, 那么 常数必须是 矩阵的 列向量 的线性组合。

Big question: what’sallowto be solved ?

一种视角： 由矩阵的列向量组构成的 张成空间，上面的点，都是 矩阵方程 有解的

矩阵方程可以视为，矩阵的列向量的排列组合，系数为的元素。

A useful approach is to chooseand find the vectorcorresponding to that solution.
The components ofare just the coeffcients in a linear combination of columns of.
The system of linear equationsis solvable exactly whenis a vector in the column space of.

使用列向量生成 张成空间 时，张成空间 的维度跟最大线性无关组有关。

Nullspace of A

The nullspace of a matrixis the collection of solutionsto equation.

The column space of the matrix in our example was a subspace of.
The nullspace ofis subspace of. To see that it’s a vector space, check that any sum or multiple of solutions tois also a solution:and

In the example:

• the nullspaceconsists of all multiples of

column 1 plus column 2 minus column 3 equals the zero vector. This nullspace is a line in

Other values of b

The solutions to the equation:

• 以上的 解空间 并不是一个 子空间（被要求 线性闭合）。
  • 零向量 并不是等式的解。
• 形成的 解集 是在三维空间中的一条经过 (1, 0, 0)、(0, -1, 1) 但没有过(0, 0, 0) 的直线。

Pivot variables， Special solutions

So this is the, turning the idea, the definition, into an algorithm

Computing the nullspace

The nullspace of a matrixis made up of the vectorfor which.

Suppose:

(Note that the columns of this matrixare not independent.)

Our algorithm for computing the nullspace of this matrix uses the method of elimination, despite the fact thatis not invertible.

使用 消去法 来计算 零空间，无需考虑 矩阵是否可逆。

(We don’t need to use an augmented matrix because the right side(the vector) isin this computation.)
不用写成 增广矩阵 的形式，因为 常数项是 零向量。

The row operations used in the method of elimination don’t change the solution toso they don’t change the nullspace. (They do affect the column space.)

消去法 的行变换，不会改变 矩阵方程的解。

The first step of elimination gives us:

• We don’t find a pivot in the second column, so our next pivot is the 2 in the third column of the second row:

我们没能在第二列找到一个主元，因此 第二主元 在第二行的第三列。

• The matrixis in echelon(staircase) form. The third row is zero because row 3 was a linear combination of rows 1 and 2; it was eliminated.

rank

The rank of a matrixequals the number of pivots it has.

In this example, the rank of(and of) is 2.

一个矩阵的秩 = 主元数量

Special solutions

Once we’ve found, we can use back-substitution to find the solutionto the equation. In our example, columns 1 and 3 are pivot columns containing pivots, and columns 2 and 4 are free columns. We can assign any value toand; we call these free variables. Supposeand. Then:

一旦我们利用 消元法 获得 梯形矩阵，便可 回代 获得矩阵方程的 解。

• 存在 主元 的列，被称为 主元列
• 其余列，被称为 自由列

在 向下消元 获得矩阵

and:

So one solution is(because the second column is just twice the first column). Any multiple of this vector is in the nullspace.

原矩阵的第二列刚好是，第一列元素的两倍。
特解的线性组合，仍然属于该 零空间。

Letting a different free variable equal 1 and setting the other free variables equal to zero gives us other vectors in the nullspace. For example:

hasand. The nullspace ofis the collection of all linear combinations of these “special solution” vectors.

• 对 矩阵自由列对应的，附上不同的值，会获得一个新的 特解 “special solution”

The rankofequals the number of pivot columns, so the number of free columns is: the number of columns(variables) minus the number of pivot columns.
This() equals the number of special solution vectors and the dimension of the nullspace.

• 矩阵的秩= 主元列数
• 自由列数
• 既是 nullspace 的维数，也是 最大线性无关组 的方程数。

rref: Reduced row echelon form

简化列阶形式

By continuing to use method of elimination we can convertto a matrixin reduced row echelon form (rref form), with pivots equal to 1 and zeros above and below the pivots.

原矩阵在使用 消元法 之后，获得的形式是 矩阵，为了获得 rref形式的矩阵需要进行进一步的化简：

1. pivots 简化为零。
2. 利用向上消元，将 主元之上 的元素消除。

By exchanging some columns,can be rewritten with a copy of the identity matrix in the upper left corner, possibly followed by some free columns on the right.
If some rows ofare linearly dependent, the lower rows of the matrixwill be filled with zeros”

(Hereis ansquare matrix.)

• 对于rref形态的矩阵，进行一系列的列变换，能够生成，左上子块为单位矩阵，又上子块为 自由矩阵 的矩阵。其中 自由矩阵为(by)的形状。

Ifis the nullspace matrixthen. [Hereis an (by) square matrix] The columns ofare special solutions.

矩阵中单位矩阵作为的填充。
矩阵的各个列向量，为 特解。

Solvability and Complete solution

这一节，原标题为 Solving: row reduced form，但是我感觉 可解性和解的结构 更到位。

When doeshave solutions, and how can we describe those solutions?

Solvability conditions on b

We again use the example:

The 3rd row ofis the sum of its first and second rows, so we know that ifthe 3rd components. Ifdoes not satisfythe system has no solution. If a combination of the rows ofgives the zero row, then the same combination of the entries ofmust equal zero.

因为 矩阵的前两行之和 刚好为第三行。由此可以得知，必须满足否则就不会有解。

One way to find out whetheris solvable is to use elimination on the augmented matrix. If a row ofis completely eliminated, so is the corresponding entry in. In our example, row 3 ofis completely eliminated:

Ifhas a solution, then b_3-b_2-b_1=0. For example, we could choose.

From an earlier lecture, we know thatis solvable exactly whenis in the column space. We have these two conditions on; In fact they are equivalent.

之前的章节中，我们已经知道，要有解，那么 向量必定在 列空间. 该条件与常数项约束条件等价。

Complete solution

在这里 note 一下，接下来的内容涉及两种解，英译中是一样的，不好区别，直接在这里标明。

• particular solution: 这个是的特解。
• special solution: 这个是的解，也是 nullspace 的一个向量。

In order to find all solutions towe first check that the equation is solvable, then find a particular solution.
We get the complete solution of the equation by adding the particular solution to all the vectors in the nullspace.

为了寻求中的所有解。

1. 检查等式的 可解性。常量是否在 列空间。
2. 寻找一个 特解。
3. 在 特解 基础上，加上对应 零空间 的所有解。

particular solution

One way to find a particular solution to the equationis to set all free variables to zero, then solve for the pivot variables.

一种寻找特解的方法：

• 将所有 自由变量 设置为 0。
• 然后求解 主变量。

For our example matrix, we letto get the system of equations:

which has the solution,. Our particular soution is:

Combined with the nullspace

The general solution tois given by, whereis a generic vector in the nullspace. To see this, we addtoand getfor every vectorin the nullspace.

Last lecture we learned that the nullspace ofis the collection of all combinations of the special solutionsand. So the complete solution to the equationsis:

Rank

The rank of a matrix equals the number of pivots of that matrix. Ifis anbymatrix of rank, we knowand

• 矩阵的秩 等于该矩阵的 主元数。
• 若矩阵的形状是秩为，我们可以知道，&&.

Full column rank

That means

If, then from the previous lecture we know that the nullspace has dimensionand contains only the zero vector. There are no free variables or special solutions.
Ifhas a solution, it is unique; there is either 0 or 1 solution. Examples like this, in which the columns are independent, are common in applications.
We know, so ifthe number of columns of the matrix is less than or equal to the number of rows. The row reduced echelon form of the matrix will look like. For any vectorinthat’s not a linear combination of the columns of, there is no solution to.

矩阵阶数等于矩阵列时，可以知道 nullspace 的维度为0 只包含 零向量。没有 自由变量 和特解。

如果有解，那么该解是唯一的。只有 0或1 个解。

Full row rank

If, then the reduced matrixhas no rows of zeros and so there are no requirements for the entries ofto satisfy. The equationis solvable for every. There arefree variables, so there arespecial solutions to

如果是行满秩矩阵 满足, 那么rref形态一定是。有个自由变量，有个的特解。

Full row and column rank

Ifis the number of pivots of, thenis an invertible square matrix andis the identity matrix. The nullspace has dimension zero, andhas a unique solution for everyin.

行列满秩矩阵，任意都有唯一的解。

Summary

Ifis in row reduced form with pivot columns first(rref), the table below summarizes our results.