2.3. Dimension, basis

₁ , ..., _n

a₁ , a₂ , ..., a_n K

a_{1 1} + a_{2 2} + ··· + a_{n n} =

and at least one a_i 0.
If

a_{1 1} + a_{2 2} + ··· + a_{n n} =    <=>   a₁ = a₂ = ··· = a_n = 0,

then the set S of vectors is linearly independent.

Example
The set { ₁, ₂, ... , _n} Rⁿ is linearly independent, since
a_{1 1} + a_{2 2} + ... + a_{n n} =
<=> a₁ ( 1, 0, ... , 0 ) + a₂ ( 0, 1, ... , 0 ) + ... + a_n ( 0, 0, ... , 1 ) = ( 0, 0, ... , 0 )
<=> ( a₁, a₂, ... , a_n ) = ( 0, 0, ... , 0 )
<=> a₁ = 0, a₂ = 0, ... , a_n = 0.

All vectors of a linearly independent set S are , because we have, for example

0·₁ + 1· + 0·₃ + ··· + 0·_n =

Thus, if    S, S is linearly dependent.
If ,    V, then {} is linearly independent. If {₁ , ..., _n } is linearly dependent, then a_{i i} = and at least one of the coefficients is 0, say, a_k . Then _k can be written as a linear combination _k = (-1 / a_k ) a_{i i} .

Example 1: A linearly independent set
Example 2: A linearly dependent set

An infinite subset of S of V is linearly independent if every finite subset of S is. Any subset of a linearly independent set is again linearly independent.

A set S V of vectors is a basis for V if it is linearly independent and spans V (L (S) = V). A vector space usually has several different bases.

Example 3: A space spanned by a set of vectors
Example 4: A subspace of R³
Example 5: A basis for R³

Proposition 4 If {₁ , ₂ , ..., _n } is a basis for V, then every vector V can be uniquely written as a linear combination of the _i 's.

Proof. Since L{₁ , ₂ , ..., _n } = V, there exist numbers a₁ , ..., a_n :

= a_{1 1} + a_{2 2} + ··· + a_{n n} .

If also

= b_{1 1} + b_{2 2} + ··· + b_{n n}

then

	= (a1 - b1 )1 + (a2 - b2 )2 + ··· + (an - bn )n
=>	a1 - b1 = a2 - b2 = ··· = an - bm = 0

and b_i = a_i   i.

The numbers a_i in the above representation of are the coordinates of relative to the basis {₁ , ..., _n }. The coordinates are given in the order of the members of a basis.

If V has a finite subset which spans it(V = L{₁ , ..., _p }), we say that V is finite dimensional.

The proof of the following result is omitted:

Proposition 5. A finite dimensional vector space V has a finite basis and any two bases of V contain the same number of vectors.

({₁, ..., _p }  and  {₁ , ..., _n }    bases for V   => n = p)

The number of vectors in a basis for V is called the dimension of V and it is denoted by dim V.

More generally, the dimension of a vector space V is dim V = sup{p N | there exist p linearly independent vectors in V}.

Example 6: The dimension of a vector space

Notice! In the sequel, the order of the vectors in a basis { ₁ , ..., _n } must not be changed!!