lecture 8 - vector spaces 2

Question 1

What is basis?

Answer 1

B = {V1, V2, .. , Vn} are basis if properties:
- vectors linearly independent
- vectors span the vector space (check against vector space principles)
They are the basis if :when multiplied by any (x,y) or (x,y,z), it produces a vector in the space

Question 2

What is steinitz lemma?

Answer 2

If you have linearly independent set of vectors (remove dependencies by cancelling), then the number of vectors in the independent set cannot exceed the number of vectors in the spanning set.
> maximum size of a linearly independent set is same as its size of a basis

Question 3

How to find the basis ?

Answer 3

if 2D, check if they are not multiples of each other
if 3D, put the vectors in a matrix and check if determinant is nonzero (non singular)
if independent and matches space’s dimension then its a basis!!

Question 4

What is dimension?

Answer 4

symbolised as dim(E).
dimension is the order of the basis.
e.g. a 3D space like R^3 will have the dimension of 3.
> max number of linearly independent vectors of E
> minimum number of vectors spanning E

Question 5

What is grassmans formula?

Answer 5

For vector subspaces (U,+, .) and (V, +, .) of (E,+, .)
Then dim(U + V) + dim (U n V) = dim(U) + dim(V)
If we sum two sets, its resulting dimension depends on each set separately and their intersection