LA 学习笔记 - Ch2 Vector Spaces
0
(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:
- linearly independent.
- 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:
- Span the space(A set of vectors)
"The columns span the column space."- Satisfy some conditions(the vectors in the space)
"The nullspace consist of all vectors that satisfy Ax = 0."
Four Fundamental Subspaces
- Column space - C(A), dimension = r.
- Nullspace - N(A), dimension = n-r. (Ax = 0的解向量)
- Row space - C(A_T), dimension = r.
- 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