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Chapter 3.4: Basis and Dimension
We say that vectors are linearly independendent if
We say that span if any can be written as a linear combination of .
Let be a vector space.
The set is called a basis in if
- are linearly independent
- They span
The number of vectors in the basis is called the dimension of
All bases in have the same number of elements called the dimension of .
Note: it is possible for a "finitely defined" Vector space to have infinite dimension.
If is a spanning set of , , then any vectors are linearly independent.
Proof. (substitution/elimination process).
If , is a 0 vector, then there is nothing to prove:
If , then . At least one of the 's is nonzero. Without loss of generality, we assume that it is which is nonzero.
Therefore , so is a spanning set for .
Repeat the process with the above spanning set and . When we can no longer continue, we end up with as a spanning set for , and
has bases , and , .
Let , , and
Note: is a spanning set of .
If and are two bases in , then .
Proof. Assume 's are spanning set and , then from Thm. 3.4.1, we know that are linearly dependent. Not a basis (contradiction)
Flip and by same logic → contradiction. Therefore .
If the vector space has no basis of finitely many vectors, we say that has infinite dimension.
If , we say that
Find the dimension of the space of solutions to the system
Three variables, two constraints:
Note that the solutions form a vector space, and the set of solutions will be the nullspace of the coefficient matrix .
The solution is of the form , so the basis has only one vector. Therefore
Find the dimension of the nullspace for the operation over
In other words, does . For a polynomial of degree ≤ 1, . If , then , and is anything. Therefore, , the basis is , and .
A basis in would be:
Let be a vector space, and . Then
- any linearly independent vectors span
- any vectors that span are linearly independent.
- assume they do not span : there is a vector which is not a linear combination of 's, then would be linearly independent. this contradicts #Theorem 3.4.1 since there are vectors with only in a basis.
- assume they are linearly dependent: then , where are not all 0. This means that one vector could be written as a linear combination of other vectors, so vectors would span . This contradicts the definition of a basis, stating that vectors must be linearly independent.
- No set of fewer than vectors spans
- Any linearly independent vectors could be extended (by more vectors) to form a basis.
- If , , span , we can pare them down to vectors, which would be a basis.