Basis and Dimension

Question 1

Explain why LI sets are never bigger than spanning sets.

Answer 1

If a vector space can be spanned by n vectors, then any LI subset of V has at most n vectors.
Or, if V has a subset of m LI vectors, then any spanning
set of V has at least m vectors.
Or, the size of any LI subset of V ≤ size of any spanning set of V

Question 2

Give the definition of a basis.

Answer 2

A set of vectors {u1, · · · , um} in V is called a basis of V if:
1. {u1, · · · , um} is LI,
2. {u1, · · · , um} spans V.
Equivalently, a basis is a LI spanning set of V, or a largest possible LI set in V, or a smallest possible spanning set of V.

Question 3

What is a fact about bases.

Answer 3

All bases have the same number of vectors in them.

Question 4

Give the definition of dimension.

Answer 4

If V has a finite basis {u1, · · · , um}, then the dimension of
V is m, i.e. the number of vectors in any basis of V:
dim(V) = m.
We also say that V is finite-dimensional.

Question 5

Give the 5 dimension theorems

Answer 5

1. For any vector space V:
   size of any LI set ≤ dim(V) ≤ size of any spanning
   set.
2. From this inequality it is implied that every LI subset of V can be extended to a basis of V and every spanning set of V can be reduced to a basis of V.
3. The zero vector space {0} has no basis, because no LI set of vectors can contain 0. Therefore, dim({0}) = 0.
4. If {u1, · · · , um} is a basis of a vector space V, then so is {a1u1, · · · , amum} for some non-zero scalars
   a1, · · · , am ∈ R.
5. Suppose we know that dim(V) = m and m < ∞. Then,
6. Any LI set {u1, · · · , um} of m vectors is a basis of V,
7. Any spanning set {u1, · · · , um} of m vectors is a basis of V.

Question 6

Define the dimension of subspaces.

Answer 6

Suppose dim(V) = n and that W is a subspace of V. Then

1. 0 ≤ dim(W ) ≤ dim(V),
2. dim(V) = dim(W ) ⇔ W=V,
3. dim(W ) = 0 ⇔ W = {0}.

Question 7

Define an ordered basis.

Answer 7

An ordered basis is the set {u1, · · · , un} with the given order of the vectors.

Question 8

Define coordinates.

Answer 8

Suppose B = {u1, · · · , un} is an ordered basis of a vector space V. Then for every v ∈ V, there exist scalars x1, · · · , xn ∈ R such that v = x1u1 + · · · + xnun, where the tuple (x1, · · · , xn) is called the coordinates of v relative to the ordered basis B.
We can use coordinates to identify n-dimensional vector spaces with R^n:
V∋v = x1u1 + · · · + xnun ←→ (x1, · · · , xn) ∈ R^n
For instance, P2 = {a+bx+cx^2
|a, b, c ∈ R} can be identified with R^3, where a + bx + cx^2 ↔ (a, b, c).