线性代数导论 (3-Vector Spaces and Subspaces)
Vector Spaces and Subspaces
Spaces of Vectors
The space $R^n$ consists of all column vectors $v$ with n components.
“Inside the vector space” means that the result stays in the space.
A subspace of a vector space is a set of vectors (including 0) that satisfies two requirments:
If $v$ and $w$ are vectors in the subspace and $c$ is any scalar, then
(i) $v$ + $w$ is in the subspace
(ii) $cv$ is in the subspace
Every subspace contains the zero vector.
The column space consists of all linear combinations of the columns.
The system $Ax=b$ is solvable if and only if $b$ is in the column space of $A$.
The Nullspace of A: Solving Ax == 0
The nullspace of $A$ consists of all solutions to $Ax = 0$. $N(A)$ is a subspace of $R^n$.
The nullspace consists of all combinations of the special solutions.
The reduced row echelon matrix $R$ has zeros above the pivots
The Rank and the Row Reduced Form
The rank of $A$ is the number of pivots.This number is $r$.
$Ax=0$ has $r$ pivots and $n-r$ free variables.N(A) contains the $n-r$ special solutions.
The Complete Solution to $Ax = b$
| rank | R | A | Ax = b solution |
|---|---|---|---|
| $r=m=n$ | $\begin{bmatrix} I \end{bmatrix}$ | Square and invertible | 1 |
| $r=m<n$ | $\begin{bmatrix} I & F \end{bmatrix}$ | Short and wide | ∞ |
| $r=n<m$ | $\begin{bmatrix} I \ 0 \end{bmatrix}$ | Tall and thin | 0 or 1 |
| $r<n,r<m$ | $\begin{bmatrix} I & F \ 0 & 0 \end{bmatrix}$ | Not full rank | 0 or ∞ |
Independence, Basis and Dimension
The columns of $A$ are independent if $x = 0$ is the only solution to $Ax = 0$.
The vectors $v_1, \ldots , v_r$ span a space if their combinations fill that space.
The sequence of vectors $v_1,\ldots,v_n$ is linearly independent if the only combination that gives the zero vector is $0v_1+0v_2+0v_3+\cdots+0v_n$.
等价语句
- elimination produces n pivots
- the inverse exists
- non-singular matrix
- The column of $A$ are independent
- the rank of $A$ equals to $n$
- There are $n$ pivots and no free variables
- Only $x=0$ is in the nullspace
The row space of $A$ is $C (A^T)$. It is the column space of $A^T$.
A basis for a vector space is a sequence of vectors with two properties: The basis vectors are linearly independent and they span the space.
The dimension of the space is the number of vectors in every basis.
Dimensions of the Four Subspaces
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