Vector Spaces and Subspaces

Question 1

What is the main goal of vector spaces?

Answer 1

The main goal is to look for non-geometric mathematical objects (so called vectors) that satisfy the same following properties as geometric vectors in n-space R^n:

1. If we add to vectors, the result is another vector,
2. We can multiply a vector by a scalar to get another vectors,
3. There exists a zero vector,
4. Every vector has a negative, which is unique.

Question 2

What is a vector space?

Answer 2

A Vector Space consists of the following:
1. A set V, to whose elements we refer as vectors,
2. An addition operation for two vectors in V (+),
3. A scalar multiplication of a vector V by a real
number c,
such that the 10 properties hold.

Question 3

What are the 10 properties of vector spaces?

Answer 3

1. If u,v∈V, then their sum u+v is also in V,
2. If u is a vector in V and c a real number, then the result of scalar multiplication cu is in V,
3. The zero vector 0∈V, such that 0+u=u, ∀u∈V,
4. For every vector u in V, there exists another vector −u in V, such that u+(−u)=0,
5. u+v=v+u, for every u,v∈V,
6. u+(v+w)=(u+v)+w, for every u,v,w∈V,
7. c(u+v)=cu+cv, for every u,v∈V and c∈R,
8. (c+d)u = cu+du, for every u∈V and c,d∈R,
9. (cd)u = c(du), for every u∈V and c,d∈R,
10. For every vector u in V, we have 1u=u.

Question 4

Describe how the 10 properties of vector spaces fit into 3 principles.

Answer 4

Properties 1-2: closure principles
Properties 3-4: existence principles
Properties 5-10: arithmetic principles

Question 5

How do we know is something is a vector space?

Answer 5

By checking all the principles (closure, existence and arithmetic
properties).

Question 6

What is a fact about vector spaces?

Answer 6

The set of all mxn matrices with real entries Mmxn(R) is a vector space for any m,n≥1.

Question 7

Define vector spaces of functions.

Answer 7

Let [a,b] denotes the interval {x ∈ R| a ≤ x ≤ b} and F[a, b] the set of all functions with domain [a, b] and values in R, i.e.
{f | f : [a, b] → R}.
For f,g∈F[a,b], f=g ⇔ f(x)=g(x), for all x∈[a,b].

Question 8

What are the operations in F[a,b]?

Answer 8

1. (f+g)(x) = f(x)+g(x), for all x∈[a,b],

2. (cf)(x) = c(f(x)), for any c∈R and x∈[a,b]

Question 9

What is a fact about vector spaces of functions?

Answer 9

F[a,b] is a vector space.

Hint: The zero function in F[a,b] is the function that sends every x∈[a,b] to zero, i.e. 0(x)=0, for all x∈[a,b]

Question 10

Define a subspace.

Answer 10

A subset W of a vector space V is called a subspace of V if it is a vector space with the same operations as in V.

Question 11

Describe the subspace test.

Answer 11

If V is a vector space and W⊂V, then W is a subspace of V if and only if:
1. 0∈W,
2. W is closed under addition: u+v∈W , for every
u,v∈W,
3. W is closed under scalar multiplication: cu∈W for
every u∈W and c∈R.

Question 12

What are 4 facts about subspaces?

Answer 12

1. For every vector space V, {0} and V are subspaces of V.
2. Any plane going trough 0 in R^3 is a subspace of R^3.
3. Any plane in R^3 that does not go through the origin is not a subspace of R^3.
4. Any line L = {tv|t ∈ R} in R^n that goes through 0 is a subspace of R^n.

Question 13

Describe the subspaces of polynomial functions.

Answer 13

The set P of all polynomial functions of the form
p(x) = a0+a1x+a2x^2+a3x^3+···+anx^n, (ai∈R, n ≥ 0)
is a subspace of the vector space of F[R], which is the vector space of all functions from R to R. Note the zero polynomial 0(x)=0, for all x∈R.

Question 14

Describe the subspaces of 3x3 symmetric matrices..

Answer 14

The set of 3×3 symmetric matrices, i.e.

S = {A∈M3×3(R) | A^T = A}, is a subspace of M3×3(R) with the same operations (addition and scalar multiplication).