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

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

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 $R^n$ consists of all column vectors $v$ with $n$ components.
空间中的任意向量的linear combination还必须在空间中。
Vector space requirements:

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

Vector spaces other than $R^n$:

Subspaces
A subspace of a vector is a set of vectors (including $0$) that satisfies: all linear combinations stay in the subspace.
All possible subspaces of $R^3$:

Two subspaces: $P$ and $L$
$P \cup L$ = all vectors in $P$ or $L$ or both: this is not a subspace
$P \cap L$ = all vectors in both $P$ and $L$: is a subspace.

Column Space of $A$

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


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