Supplement: Basis, Dimension and Rank
1. Basis
Definition of basis:
The vectors in a vector space V are said to form a basis of V if
(a) span V (i.e., ).
(b) are linearly independent.
Example:
. Are and a basis in ?
[solution:]
and form a basis in since
(a) (see the example in the previous section).
(b) and are linearly independent (also see the example in the previous section).
Example:
. Are and a basis in ?
[solution:]
and are not a basis of since and are linearly dependent,
.
Note that .
Example:
. Are and a basis in ?
[solution:]
andare not a basis in since and are linearly independent,
.
Example:
Let
.
Are S a basis in ?
[solution:]
(a)
For any vector , there exist real numbers such that
.
we need to solve for the linear system
.
The solution is
.
Thus,
.
That is, every vector in can be a linear combination of and .
(b) Since
,
are linearly independent.
By (a) and (b), are a basis of .
Important result:
If is a basis for a vector space V, then every vector in V can be written in an unique way as a linear combination of the vectors in S.
Example:
. S is a basis of . Then, for any vector ,
is uniquely determined.
Important result:
Let be a set of nonzero vectors in a vector space V and let . Then, some subset of S is a basis of W.
Important result:
Let be a basis for a vector space V and let is a linear independentset of vectors in V. Then, .
Corollary:
Let and be two bases for a vector space V. Then, .
Note:
For a vector space V, there are infinite bases. But the number of vectors in two different bases are the same.
Example:
For the vector space ,
is a basis for (see the previous example). Also,
is basis for .
There are 3 vectors in both S and T.
2. Dimension
Definition of dimension:
The dimension of a vector space V is the number of vectors in a basis for V.
Example:
is basis for .
The dimension of is 3.
Important result:
Let V be an n-dimensional vector space, and let be a set of n vectors in V.
(a)If S is linearly independent, then S is a basis for V.
(b)If S spans V, then S is a basis for V.
Example:
Is a basis for ?
[solutions:]
Since is a 3-dimensional vector space, not like in the previous example, we only need to examine whether S is linearly independent or S spans . We don’t need to examine S being both linearly independent and spans V.
Example:
Is a basis for ?
[solutions:]
Since is a 3-dimensional vector space, we only need to examine whether S is linearly independent or S spans . Because
,
are linearly independent. Therefore, are a basis of
3. Rank of a Matrix:
Recall:et
.
The i’th row of A is
,
and the j’th column of A is
Definition of row space and column space:
,
which is a vector space under standard matrix addition and scalar multiplication, is referred to asthe row space. Similarly,
,
which is also a vector space under standard matrix addition and scalar multiplication, is referred to as the column space.
Definition of row equivalence:
A matrix B is row equivalent to a matrix A if B result from A via elementary row operations.
Example:
Let
Since
,
,
,
are all row equivalent to .
Important Result:
If A and B are two row equivalent matrices, then the row spaces of A and B are equal.
Definition of row rank and column rank:
The dimension of the row space of A is called the row rank of A and the dimension of the column space of A is called the column rank of A.
Example (continue):
Let
Since the basis of the row space of A is
,
the dimension of the row space is 3 and the row rank of A is 3. Similarly,
is the basis of the column space of A. Thus, the dimension of the column space is 3 and the column rank of A is 3.
Important Result:
The row rank and column rank of the matrix A are equal.
Definition of the rank of a matrix:
Since the row rank and the column rank of a matrix A are equal, we only refer to the rank of A and write .
Important Result:
Let A be an matrix.
A is nonsingular if and only if .
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