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 .
The space
Rn
R
n
consists of all column vectors
v
v
with 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 :
Two subspaces:
P
P
and
P∪L
P
∪
L
= all vectors in
P
P
or or both: this is not a subspace
P∩L
P
∩
L
= all vectors in both
P
P
and : is a subspace.
Column Space of A A
Column space : all linear combinations of column vectors of matrix
A
A
.
The system is solvable if and only if
b
b
is in the column space of .
The subspace
SS
S
S
is the span of
S
S
, containing all combinations of vectors in .