MIT18.06学习笔记 - Lecture 6: Vector spaces and subspaces

Course Name:MIT 18.06 Linear Algebra
Text Book: Introduction to Linear Algebra
章节内容: 3.1


课程提纲
1. Vector spaces and subspaces
2. Column Space of A A

课程重点

Vector spaces and subspaces

Three level of understanding: numbers, vectors and vector spaces.
Without seeing vector spaces and especially their subspaces, you haven’t understood everything about Ax=b.
The space Rn R n consists of all column vectors v v with n components.
空间中的任意向量的linear combination还必须在空间中。
Vector space requirements:

  • v+w v + w and cv c v are in the space
  • all combinations cv+dw c v + d w are in the space
  • Every vector space must have its own zero vector.


Vector spaces other than Rn R n :

Subspaces
A subspace of a vector is a set of vectors (including 0 0 ) that satisfies: all linear combinations stay in the subspace.
All possible subspaces of R3:

Two subspaces: P P and L
PL P ∪ L = all vectors in P P or L or both: this is not a subspace
PL P ∩ L = all vectors in both P P and L: is a subspace.

Column Space of A A

Column space C(A): all linear combinations of column vectors of matrix A A .
The system Ax=b is solvable if and only if b b is in the column space of A.


The subspace SS S S is the span of S S , containing all combinations of vectors in S.

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