Summarized from ‘Introduction to Linear algebra’ from Gilbert Strang.

## Definition of The Vector Spaces

**Vector space** is a very important concept, which is denoted by $R^1, R^2, R^3 … R^n$, which consists of a *a whole collection of vectors*. For example, $R^5$ contains all column vectors with five components, the so-called “ive-dimentional space”.

DEFINITIONThe space $R^n$ consists of all column vectors v with n components.

Here are three vector spaces other than $R^n$:

MThe vector space ofall real 2 by 2 matrices

FThe vector space of*all real functions $f(x)$

ZThe vector sapce that consists only of azero vector

**PS***:The function space F is infinite-dimensional. A smaller function space is $P$, or $P_n$ containing all polynomials $a_0 + a_1x … a_nx^n$ of degree n.

## Subspaces

A plane in three-dimensional space is not $R^2$ (even if it looks like $R^2$). The vectors have three components and they belong to $R^3$. The plane is a vector space ** inside $R^3$**.

DEFINITIONA subspace of a vector space is a set of vectors (including 0) that satisfies two requirements:( i ) $v+w$ is in the subspaceIf $v$ and $w$ are vectors in the subspace and $c$ is any scalar, then

( ii ) $cv$ is in the subspace

There are two facts,

- Every subspace contains the zero vector.
- Lines through the origin are also subspaces.

( $L$ ) Any line through (0, 0, 0)

( $P$ ) Any plane through (0, 0, 0)

( $R^3$ ) The whole space

( $Z$ ) The single vector (0, 0, 0)

## The Column Space of A

Start with the columns of A and ** take all their linear combinations. This produces the column space of A, **. $C(A)$ contains not just the $n$ columns of $A$, but all their combinations $Ax$.

`It is a vector space made up of column vectors`

DEFINITION The

consists ofcolumn space. The combinations are all possible vectors $Ax$. They fill the column space $C(A)$.all linear combinations of the columns

To solve $Ax = b$ is to express b as a combination of the columns.

The system $Ax = b$ is solvable if and only if b is in the column space of $A$.

**Important:** Instead of columns in $R^m$, we could start with any set $S$ of vectors in a vector space $V$. To get a subspace $SS$ of $V$, we take all combinations of the vectors in that set:

$S$ = set of vectors in V (probabley not a subspace)

$SS$ = all combinations of vectors in $S$

$SS$ = all $c_1v_1 + …+ c_Nv_N$ = the subspace of $V$ “spannd” by $S$.

When $S$ is the set of columns, $SS$ is the column space. When there is noly one nonzero vector $v$ in $S$, the subspace $SS$ is the line through $v$. ALways $SS$ is the smallest subspace containing $S$.

The subspace $SS$ is the “span” of $S$, containing all combinaitons of vectors in $S$.

## Some Problems to Learn

In the definition of a vector space, vector spaces are not necessarily column vectors. In the definition of a vector space, vector addition $x+y$ and scalar multiplicaiton $cx$ must obey the following eight rules:

- $x+y=y+x$
- $x+(y+z)=(x+y)+z$
- There is a unique ‘zero vector’ such that $x+0=x$ for all $x$
- For each $x$ there is a unique vector $-x$ such that $x + (-x) = 0$
- 1 times $x$ equals $x$
- $(c_1c_2)x=c_1(c_2x)$
- $c(x+y)=cx + cy$
- $(c_1+c_2)x = c_1x + c_2x$

For example:

Problems,

- The possitive numbers with $x + y$ and $cx$ redefined to equal the usual $xy$ and $x^c$ do satisfy the eight rules. Test rule 7 when $c =3, x = 2, y =1$. (Then $x+y=2$ and $cx=8$.) Which number acts as the “zero vector”?
$c(x+y)\to(xy)^c$, and zero vector is 1

- $A$ =
$$
\begin{bmatrix}
1 & 0

0 & 0

\end{bmatrix} $$ and $B$ = $$ \begin{bmatrix} 0 & 0

0 & -1

\end{bmatrix} $$ , if a subspace of M does contain $A$ and $B$, must it contain $I$?From the DIFINITION of subspace, we can find when $u$ and $w$ are two vectors in the subspace. $u + w$ is also in the subspace is also in the subspace. Because that $A - B$ = $$ \begin{bmatrix} 1 & 0

0 & 1

\end{bmatrix} $$ = $I$, this subspace must contain $I$. - The subspaces of $R^3$ are planes, lines , $R^3$ itself, or $Z$ containing only (0, 0 ,0).
- Show that the set of singular matrices in $M$ is not a subspace.
Import two singular matrices, $A$ = $$ \begin{bmatrix} 1&0

0&0

\end{bmatrix} $$ , and $B$ = $$ \begin{bmatrix} 0&0

0&1

\end{bmatrix} $$ . It’s because that the sum of $A,B$ is $$ \begin{bmatrix} 1&0

0&1

\end{bmatrix} $$ which is invertible, so the set of singular matrices in $M$ is not a subspace. - The columns of $AB$ are combinations of the columns of $A$. This means: The column space of $AB$ is contained in (possibly equal to) the column space of A. Give an example where the column spaces of $A$ and $AB$ are not equal.
If $B$ is a zero matrix, while $A!=0$, then $AB$ is smaller than $A$ and not equal to each other.

- If $A$ is any n by n invertible matrix, then its column space is $R^n$, WHY?
$Ax = b$ is always solved by $x = A^{-1}b$