Span of Vectors in a Vector Space

Question 1

Describe the 5 important vector spaces and subspaces.

Answer 1

1. R^n: with standard vector addition and scalar multiplication.
2. M2×2(R) =
   {[a b]
   [c d] | a,b,c,d are real numbers}: with entry-wise addition and scalar multiplication.
3. F[R] = {all functions F : R→R}: with point-wise addition and scalar multiplication.
4. The set P of all polynomial functions of the form
   p(x) = a0+a1x+a2x^2+a3x^3+···+an^x, (ai ∈ R, n ≥ 0) is a subspace of the vector space of F[R].
5. The set of 3×3 symmetric matrices, i.e.
   S = {A∈M3×3(R) | A^T=A}, is a subspace of M3x3(R)

Question 2

What is the goal of span of vectors in a vector space?

Answer 2

All subspaces (except {0}) contain infinite number of vectors, but the goal is to completely identify a subspace by just a list of finite number of vectors.

Question 3

Demonstrate how to find the span of

W = {(x,y,z)∈R^3 | x−2y+z=0}.

Answer 3

W = {(x,y,z)∈R^3 | x−2y+z=0}
= {(x,y,z)∈R^3 | x=2y−z}
= {(2y−z,y,z)| y,z∈R}
= {(2y,y,0)+(−z,0,z)| y,z∈R}
= {y(2,1,0)+z(−1,0,1)| y,z∈R}
In words, W is the set of all linear combinations of the vectors
(2, 1, 0) and (−1, 0, 1), or
W = Span {(2, 1, 0),(−1, 0, 1)}.

Question 4

Demonstrate how to find the span of
S2×2(R) =
{[a b]
[b d] | a,b,d∈R}

Answer 4

S2×2(R) = 
{[a b]
 [b d]  | a,b,d∈R} =
{[a 0]   [0 b]   [0 0]
 [0 0] +[b 0] +[0 d] | a,b,d∈R} =
{a[1 0] + b[0 1] + d[0 0]
   [00]      [1 0]      [0  1] 
| a,b,d∈R}

= Span {[1 0] , [0 1] , [0 0]
             [00]   [1 0]   [0  1] 
where S2×2(R) is the set of all linear combinations of
[1 0] , [0 1] , [0 0]
[00]   [1 0]   [0  1]

Question 5

Describe Span with 3 points.

Answer 5

1. If u1,···,um are vectors in the vector space V and
   a1···,am∈R, then
   a1u1+a2u2+···+amum
   is called a linear combination of u1,···,um.
2. The set of all linear combinations of u1,···,um is
   called the span of u1,···,um:

Span{u1,···,um} = {a1u1+···+amum | a1,···,am∈R}.
The set {u1,···,um} is called the spanning set.
3. A vector space V is spanned by u1,···,um∈V if
V = Span{u1,···,um}.

Question 6

Describe the fact that spanning sets are subsets.

Answer 6

Let V be a vector space. If {u1,···,um}⊂V, then
1. U = Span{u1,···,um} is always a subspace of V.
2. If W is any subspace of V containing all vectors u1,···,um, then U⊂W . In fact, U is the smallest
subspace that contains u1,···,um.

Question 7

Give two examples of spanning sets.

Answer 7

A line L={tv | t∈R} going through the origin in R^n is a subspace of R^n spanned by v, i.e. L = Span{v}.
2. W = Span{sin(x), cos(x)} is a subspace of F[R].

Question 8

What are the two subspaces of R?

Answer 8

{0} and R

Question 9

What are the three subspaces of R^2?

Answer 9

1. The zero subspace, i.e. the origin {(0, 0)},
2. Every line going through the origin (0, 0),
3. All of R^2

Question 10

What are the four subspaces of R^3?

Answer 10

1. The zero subspace, i.e. the origin {(0, 0, 0)},
2. Every line going through the origin (0, 0, 0),
3. Every plane going through the origin (0, 0, 0),
4. All of R^3

Question 11

What are the two main problems with spans?

Answer 11

1. It is not easy to tell if two subspaces are equal based on the spanning sets. For every vector space (except {0}), there are infinitely many spanning sets (spanning sets are non-unique).
2. Having more vectors in the spanning set does not imply that the subspace they span is larger (does not imply more vectors
   in their span)

Question 12

What are 4 facts about spans?

Answer 12

1. Span{0} = 0;
2. If v≠0, Span{v} is the line through 0 with direction v;
3. If u,v≠0 and u and v are non-parallel (non-collinear), then Span{u, v} = R^2
4. If u,v,w≠0 and u, v and w are not coplanar, then
   Span{u,v,w} = R^3