Vector Spaces

Question 1

What is the definition of a vector space?

Answer 1

A vector space over a field F is a NON-EMPTY set, V , on which both VECTOR ADDITION and SCALAR MULTIPLICATION are defined

Question 2

What are the 8 axioms of vector spaces?

Answer 2

1. u+v=v+u
2. (u+v)+w=u+(v+w)
3. There exists 0 in V such that 0+u=u
4. For every u there is -u such that u+(-u)=0
5. α(u+v)=αu+αv
6. (α+β)u=αu+βu
7. (αβ)u=α(βu)
8. 1u=u

Question 3

How do you show something is a vector space over the field R^2?

Answer 3

• Define two vectors in R^2 (a1,a2) (b1,b2)

- Show that all 8 axioms hold

Question 4

How would you show something is not a vector space?

Answer 4

• Define two vectors in the vector space

- Find one axiom that does not hold

Question 5

What is the definition of a subspace?

Answer 5

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

Question 6

True or False:

The solution set of a system is a subspace if the system is homogeneous

Answer 6

True

Question 7

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

Answer 7

False

Question 8

What is the definition of a linear combination?

Answer 8

A vector v is a linear combination if:

V =α1v1+α2v2+…+αkvk

Question 9

What is the definition of a span?

Answer 9

A span is the set of all linear combinations:

Span(v1…vk) = {α1v1+…+αkvk: α1,…,αk ε F}

Question 10

What can we say about the row space and row operations of a matrix if two matrices are row equivalent

Answer 10

If two matrices are row equivalent then their row spaces are equal
Hence the row space of a matrix is invariant under row operations

Question 11

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

Answer 11

True