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(This is my notes for G. Strang. Linear Algebra and its application.)

2.1

The solution to Ax = 0: from the nullspace of A. : the null space is a Line.

Vector Space: we can take linear combination.

DEFINITION: A subspace of a vector space is a nonempty subset that satisfy the requirement for a vector space: Linear combination stay in the subspace(x + y, cx)
*重点是：进行线性操作后得到的向量 - 依然满足自己本身的性质! （始终服从自己的性质）
（Solution:
Let x1, x2 ∈ S.
Prove: (1) x1+x2 ∈ S;）
(2) c * x1 ∈ S; (take care when c = negative!))

To find C(A) and N(A):

• C(A) = all attainable right-hand side b;
• N(A) = all solution to Ax = 0.
  (p73)

2.2

Nullspace contains all combination of spacial solutions.

• conlums w/ pivot: Pivot variable;
• conlums w/o pivot: Free variable;
  pivot variables are determined by free var.

---

x_nullspace(Ax = 0)

Steps (p80) -

• Rx = 0; (indentifying pivot/free variable)
• Give ONE free var = 1, OTHER free var = 0, solve Rx = 0 and we get one special sol.
• Every free var. produces its special sol.
• The combination of special sol form nullspace - all solution to Ax = 0.

---

X_complete = x_particular + x_nullspace

(x_particular - setting all the free var. to zero.)

Steps: (p83)

2.3 Subspace and Basis

Subspace的两个条件：

• x1 + x2 属于
• c*x1 属于
  （记忆：x+y, cx）(c - any scalar)(quadrant象限不是subspace )
  zero vector will belong to every subspace.(take c = 0)

2A Ax = b: solvable iff (p71)
b can be expressed as a combination of the column of A. (b is in the column space. )

Basis for a Vector Space

A Basis for a Vector Space 满足两个条件:

1. linearly independent.
2. Span the space.

Every vector in the space = a combination of the basis  (only one way)

Dimension of a space = number of vectors in every basis.
空间的维度 就是向量基的个数。
http://www.math.uconn.edu/~troby/Math2210F09/LT/sec4_5.pdf

eg. The dimension of the column space = the **rank** of the matrix [r]
Among those [r] vectors, each vector is [1xm], therefore the column space belongs to R^m.

2.4 Four Fundamental Subspaces

定义subspace时的一些描述：

1. Span the space(A set of vectors)
   "The columns span the column space."
2. Satisfy some conditions(the vectors in the space)
   "The nullspace consist of all vectors that satisfy Ax = 0."

Four Fundamental Subspaces

1. Column space - C(A), dimension = r.
2. Nullspace - N(A), dimension = n-r. (Ax = 0的解向量)
3. Row space - C(A_T), dimension = r.
4. Left nullspace - N(A_T), dimension = m-r. (The nullspace of A_T; (A^T)*y = 0的解向量

(A --> m x n,
then $A^T$ --> n x m and
A^T's nullspace(A's nullspace) is R^m )

注意：
所有向量都是列向量。(column vectors.)

four fundamental subspace:
[线代随笔03-矩阵的四个线性子空间](
http://bourneli.github.io/linear-algebra/2016/02/28/linear-algebra-03-the-four-subspaces-of-matrix.html )

>例子答案：
第1列     第2列     第3列   
1.0000      -0.0000     0.0000   
-2.0000     1.0000      0.0000   
-5.0000     0.0000      1.0000 

2.5 Linear Transformations

Rotation
Projection
Reflection