• Support PF! Buy your school textbooks, materials and every day products Here!

Proving a set of matrices is NOT a vector space

  • #1
2
0

Homework Statement


Show that the following is NOT a vector space:
{(a, 1) | a, b, c, ∈ ℝ}
{(b, c)

Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it.

2. The attempt at a solution
I am self-studying linear algebra, and have had a difficulty conceptualizing/interpreting what a vector space actually is. However, to show that the set of all matrices of the form above is NOT a vector space, I subjected it to the addition/multiplication conditions. For example, the condition v+w ∈ V:
(a, 1) + (p, 1) = (a+p, 2)
(b, c) + (q, r) = (b+q, c+r)

The sum of the two matrices yields a matrix NOT of the original form, with 1 in the a_1,2 position. Does this prove that the set is not a vector space? Have I correctly interpreted the purpose of this problem as: "Show that one of the vector space conditions does not give back a matrix with the same form as the original matrix (a, 1, b, c)"?
Another way I interpreted the question was: "Find a way to make a, b, or c NOT a real number." However, I could not think of a way to transform them into an imaginary number without using a scalar r ∉ ℝ.

My book gives many examples of what vector spaces are, or can be, but leaves it up to the reader to infer what does not constitute a vector space.

So, have I correctly solved the problem? Any helpful tips would be greatly appreciated! No answers, please. :)

Thanks a lot,
Jordan
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,522
737

Homework Statement


Show that the following is NOT a vector space:
{(a, 1) | a, b, c, ∈ ℝ}
{(b, c)

Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it.

2. The attempt at a solution
I am self-studying linear algebra, and have had a difficulty conceptualizing/interpreting what a vector space actually is. However, to show that the set of all matrices of the form above is NOT a vector space, I subjected it to the addition/multiplication conditions. For example, the condition v+w ∈ V:
(a, 1) + (p, 1) = (a+p, 2)
(b, c) + (q, r) = (b+q, c+r)

The sum of the two matrices yields a matrix NOT of the original form, with 1 in the a_1,2 position. Does this prove that the set is not a vector space?
Yes. That's all you needed. Even just the first equation was enough.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement


Show that the following is NOT a vector space:
{(a, 1) | a, b, c, ∈ ℝ}
{(b, c)

Note: this is is meant to be a 2x2 matrix. This may not have been clear in how I formatted it.

2. The attempt at a solution
I am self-studying linear algebra, and have had a difficulty conceptualizing/interpreting what a vector space actually is. However, to show that the set of all matrices of the form above is NOT a vector space, I subjected it to the addition/multiplication conditions. For example, the condition v+w ∈ V:
(a, 1) + (p, 1) = (a+p, 2)
(b, c) + (q, r) = (b+q, c+r)

The sum of the two matrices yields a matrix NOT of the original form, with 1 in the a_1,2 position. Does this prove that the set is not a vector space? Have I correctly interpreted the purpose of this problem as: "Show that one of the vector space conditions does not give back a matrix with the same form as the original matrix (a, 1, b, c)"?
Another way I interpreted the question was: "Find a way to make a, b, or c NOT a real number." However, I could not think of a way to transform them into an imaginary number without using a scalar r ∉ ℝ.

My book gives many examples of what vector spaces are, or can be, but leaves it up to the reader to infer what does not constitute a vector space.

So, have I correctly solved the problem? Any helpful tips would be greatly appreciated! No answers, please. :)

Thanks a lot,
Jordan
Yes, your argument is correct. Also: a vector space must contain the zero vector, which your set of matrices does not.
 

Related Threads on Proving a set of matrices is NOT a vector space

Replies
6
Views
4K
  • Last Post
Replies
2
Views
774
Replies
3
Views
7K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
994
  • Last Post
Replies
2
Views
5K
Replies
14
Views
2K
Top