What is the definition of a vector space?
A vector space over a field F is a NON-EMPTY set, V , on which both VECTOR ADDITION and SCALAR MULTIPLICATION are defined
What are the 8 axioms of vector spaces?
- u+v=v+u
- (u+v)+w=u+(v+w)
- There exists 0 in V such that 0+u=u
- For every u there is -u such that u+(-u)=0
- α(u+v)=αu+αv
- (α+β)u=αu+βu
- (αβ)u=α(βu)
- 1u=u
How do you show something is a vector space over the field R^2?
- Define two vectors in R^2 (a1,a2) (b1,b2)
- Show that all 8 axioms hold
How would you show something is not a vector space?
- Define two vectors in the vector space
- Find one axiom that does not hold
What is the definition of a subspace?
A NON-EMPTY set W in V is a subspace of V if W is a vector space under the same operations
Hence it will be closed under addition and scalar multiplication
True or False:
The solution set of a system is a subspace if the system is homogeneous
True
True or False:
Even if the subspace U, W are in V; then the intersection of U and W may not be a subspace of V
False
What is the definition of a linear combination?
A vector v is a linear combination if:
V =α1v1+α2v2+…+αkvk
What is the definition of a span?
A span is the set of all linear combinations:
Span(v1…vk) = {α1v1+…+αkvk: α1,…,αk ε F}
What can we say about the row space and row operations of a matrix if two matrices are row equivalent
If two matrices are row equivalent then their row spaces are equal
Hence the row space of a matrix is invariant under row operations
True or False:
If the REF is row equivalent to A, then the row space of A is the entire field F to power n
True
