5. Vector Spaces Flashcards
Which two major properties are required for something to qualify as a vector space?
- It has to be closed under scalar multiplication
- It has to be closed under addition.
Can a set of functions from some set to R be thought of as a vector space?
Yes, because the necessary operations between those functions are defined.
What is a “linear functional”?
A function from a vector space to a real number that is linear.
What is the set of all linear functionals on some vector space “V” called?
The “dual space of V”, which is denoted V*
It is a vector space as well.
Give an example of a linear functional.
fa*(v) = a•v
Where a is a certain vector for each functional f*.
What is the “span” of a set of vectors?
The set of all linear combinations of those vectors, which is by definition a vector space.
What is a “basis” for a vector space?
A linearly independent set of vectors whose span is the vector space.
What is the “dimension” of a vector space?
The number of vectors in a basis for the vector space (this is identical no matter which basis you have; it depends on the space only)
What is the dimension of the trivial vector space V = {Ø}?
Zero, because its basis is the empty set. Its basis cannot be zero because any set with zero in it is linearly dependent.
How would you find a basis for the vector space defined by x + y + z = 0
Write out the general vector that satisfies that equation, and then write it in terms of the independent (free) variables.
What is the “Kronecker Delta”?
A function of two indices that returns 1 only if the indices are the same.
What is the standard way to produce a basis for the dual space of a vector space?
Define your basal functionals using the Kronecker delta, so that the ith-functional returns the ith-component of the given vector. Then any functional can be written as a linear combination of these basal functionals. This is called the “Dual Basis”.
How is the dimension of a vector space and the dimension of its dual space related?
They are always equal.
