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A vector space automatically has two subspaces:
When and , then is called a proper subspace.
Null Space of Matrix
Let be a matrix, and let be the set of all solutions to the system :
is clalled the null space (also called nullspace or kernel) of .
is a vector space since and
Gauss-Jordan reduction of gives , so the solutions of are
Linear Combinations and Span
The sum , where , is called a linear combination of .
The set of all such linear combinations is called the span of :
and are in
The span of and is the set of all vectors .
These two vectors form the standard basis in 2D space.
is a subspace of
The set is a spanning set for if
If a subset of is a spanning set of , then itself is a spanning set of (set linear coefficient of other terms in set to 0).
These are the standard basis vectors for 3D space.
Is a spanning set of ?
This has a solution for any , so it is indeed a spanning set (the vectors in the original problem are noncoplanar, so that's easier to visualize)
is the set of polynomials of degree < 3.
Given , , and ,
This is true since
The set is called linearly dependent
- If span and one of these vectors can be written as a linear combination of the others, then those vectors span .
- Given vectors , it is possible to write one of the vectors as a linear combination of the other vectors iff there exist scalars (not all zero!) such that .