Basis and Dimension
Given a vector space \( V \), a basis is a list of vectors \( B = \{ u_1, u_2, \ldots, u_n \} \) that satisfies:
- each \( u_i \in V \)
- \( B \) is lin. ind., ie. \[ c_1 u_1 + c_2 u_2 + \cdots + c_n u_n = 0 \Rightarrow c_1 = c_2 = \cdots = c_n = 0 \]
- \( B \) spans / fills the whole \( V \), ie. for any \( \vec{v} \in V \), there are \( c_1, c_2, \ldots c_n \) s.t. \[ c_1 u_1 + c_2 u_2 + \cdots + c_n u_n = v \]
Basis and Dimension
Often (or in fact, always),
- Checking for lin. ind. is equivalent to solving a homogeneous system: \[ AX = 0 \]
- Checking for span is equivalent to solving an inhomogeneous system: \[ AX = \vec{b} \] for all possible \( \vec{b} \)
(You do need to find \( A, \vec{b} \) for each problem.)
Basis and Dimension
The dimension of \( V \), \( \dim(V) \) is defined as the number of vectors in a (any) basis.
Basis and Dimension
The dimension of \( V \), \( \dim(V) \) is defined as the number of vectors in a (any) basis.
e.g.
\[ \mathbb{R}^2 = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} \]lin ind. & span
Hence
\[ \dim(\mathbb{R}^2) = 2 \]Basis and Dimension
The dimension of \( V \), \( \dim(V) \) is defined as the number of vectors in a (any) basis.
e.g.
\[ \mathbb{R}^2 = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} \]lin ind. & span
Hence
\[ \dim(\mathbb{R}^2) = 2 \]Note:
There can be many different basis but the dimension is always the same.
Basis and Dimension
The dimension of \( V \), \( \dim(V) \) is defined as the number of vectors in a (any) basis.
e.g.
\[ \mathbb{R}^2 = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} \]lin ind. & span
Hence
\[ \dim(\mathbb{R}^2) = 2 \]e.g.
\[ \mathbb{R}^2 = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right\} \](also a basis as it is lin ind. and it spans \( \mathbb{R}^2 \))
Properties of \( \dim(V) \)
Let \( \dim(V) = n \). Then
Any list of \( m > n \) vectors must be linearly dependent, i.e.
\( n \) is the maximum number of lin ind. vectors
Pf : Any linear system with more unknowns than equations must have at least one free var. More Unknowns Theorem.
Properties of \( \dim(V) \)
Let \( \dim(V) = n \). Then
Note
Any list of \( m > n \) vectors must be linearly dependent, i.e.
\( n \) is the maximum number of lin ind. vectors
Any 3 or more vectors in \( \mathbb{R}^2 \) must be lin dep
Any 4 or more vectors in \( \mathbb{R}^3 \) must be lin dep
Any 5 or more vectors in \( \mathbb{R}^4 \) must be lin dep
Properties of \( \dim(V) \)
Let \( \dim(V) = n \). Then
Note
Any list of \( m < n \) vectors cannot span \( V \), i.e.
\( n \) is the minimum number of vectors that can span \( V \).
Pf : Given \( Ax = \vec{b} \), with # of rows of \( A \) > # of cols of \( A \), there is a \( \vec{b} \) such that \( Ax = b \) is not solvable. More Equations Theorem.
Properties of \( \dim(V) \)
Let \( \dim(V) = n \). Then
Note
Any list of \( m < n \) vectors cannot span \( V \), i.e.
\( n \) is the minimum number of vectors that can span \( V \).
Any 2 or fewer vectors cannot span \( \mathbb{R}^3 \)
Any 3 or fewer vectors cannot span \( \mathbb{R}^4 \)
Any 4 or fewer vectors cannot span \( \mathbb{R}^5 \)
Properties of \( \dim(V) \)
- The \( \dim(V) \) is the maximum number of vectors that can be lin. ind.
- The \( \dim(V) \) is the minimum number of vectors that can span the whole \( V \).
- (Suppose \( \dim V = n \)) Any \( n \) vectors that span \( V \) must be lin. ind.
- (Suppose \( \dim V = n \)) Any \( n \) vectors that are lin ind. must span.
Properties of \( \dim(V) \)
(1) or (2):
Note
\( \dim(V) \) is a unique number (basis not)
(3) and (4):
Note
In an \( n \)-dimensional vector space, any \( n \) vectors, lin ind \( \iff \) span \( \iff \) basis
Note
The dimension gives the effective degree of freedom in a vector space.
Standard Basis
\( \mathbb{R}^2 = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\} \), where the first vector is \( e_1 \) and the second is \( e_2 \). \( \dim(\mathbb{R}^2) = 2 \)
\[ \begin{pmatrix} x \\ y \end{pmatrix} = x \begin{pmatrix} 1 \\ 0 \end{pmatrix} + y \begin{pmatrix} 0 \\ 1 \end{pmatrix} = x e_1 + y e_2 \]\( \mathbb{R}^n = \operatorname{Span}\left\{ \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix}, \dots, \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix} \right\} \), where these vectors are \( e_1, e_2, \dots, e_n \). \( \dim(\mathbb{R}^n) = n \)
\[ \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = x_1 \begin{pmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{pmatrix} + x_2 \begin{pmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{pmatrix} + \cdots + x_n \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix} = x_1 e_1 + \cdots + x_n e_n \]Standard Basis
\[ \begin{gathered} M^{2\times 2} = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \mathbb{R} \right\} \\[0.5em] = \left\{ a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} + b\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} + c\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} + d\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\} \\[0.5em] = \operatorname{Span}\left\{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right\} \end{gathered} \]standard basis
\[ \dim\left( M^{2\times 2} \right) = 4 = (2\times 2) \]"Standard" Basis
\[ \begin{gathered} M^{m\times n} = \left\{ \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & & \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} : a_{ij} \in \mathbb{R} \right\} \\[0.5em] = \operatorname{Span}\left\{ \begin{pmatrix} 1 & 0 & \cdots & 0 \\ & 0\text{'s} & & \end{pmatrix}, \begin{pmatrix} 0 & 1 & \cdots & \\ & 0\text{'s} & & \end{pmatrix}, \dots, \begin{pmatrix} 0 & \cdots & 0 \\ & 0\text{'s} & \\ 0 & \cdots & 0 & 1 \end{pmatrix} \right\} \\[0.5em] \dim\left( M^{m\times n} \right) = mn \end{gathered} \]"Standard" Basis
\[ \begin{gathered} P^n = \left\{ p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \right\} \\[0.5em] = \operatorname{Span}\left\{ 1, x, x^2, \dots, x^n \right\} \end{gathered} \]"Standard" Basis
\[ \begin{gathered} P^n = \left\{ p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \right\} \\[0.5em] = \operatorname{Span}\left\{ 1, x, x^2, \dots, x^n \right\} \end{gathered} \]\( a_0 = a_0 \cdot 1 \), \( 1 \) is a polynomial of deg. 0.
Note
\[ \dim\left( P^n \right) = n + 1 \]