Linear Algebra - Vector Spaces

In summary, the first question asks for a proof that if a scalar is multiplied by the zero vector, the result is also zero. The second question asks for a proof that if a vector is multiplied by the zero vector, the result is also zero. The first and second questions are not general enough, so the student is confused. The student needs to use the axioms of vector spaces to prove the equations. The axioms include that (for a, b scalars and v a vector) (ab)v=a(bv) and v=0. The student needs to show that for any vector u, u+v=v+u=u(0).
  • #1
Delta-One
5
0
Hi,
I'm having trouble with these homework questions.

I have to prove that B*0v = 0v , where B is a scalar.

Also, I have to prove that if aX = 0v , then either a = 0 or X = 0
---where a is a scalar and X is a vector.


I know that I have to use the 8 axioms but I'm not sure where to begin.
 
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  • #2
One of the axioms is that anything times a 0 vector is 0, unless that is the one you are trying to prove.

The zero vector will contain all zero components, therefore any scalar multiplied by 0 will also be 0 as 0*a is always zero for any scalar. For your first question, the proof depends on what "B" represents.
 
  • #3
Thanks for answering the first question, but any ideas about the second (aX = 0)? These vector spaces really have me confused.
 
  • #4
No, "0 times anything is 0" is never an axiom- it's too easy to prove!
And I don't like the idea of using components to prove this- not general enough.

What is B*(u+ 0v)? (u is a vector, 0v is the 0 vector)
What is u+ 0v?

If a is not 0, what is (aX)/a? If aX= 0 what does the answer to my question tell you?
 
  • #5
I'm really sorry but I still don't see how to show that
B*0v = 0v
--where B is a constant.

Are you suggesting that I add the vector u to each side of the eqn?
 
  • #6
Delta-One said:
I'm really sorry but I still don't see how to show that
B*0v = 0v
--where B is a constant.

Are you suggesting that I add the vector u to each side of the eqn?
what are your axioms?
One possible set includes
(for a, b scalars and v a vector)
(ab)v=a(bv)

to show for some vector v
v=0
show that for any vector u
u+v=v+u=u
 

1. What is a vector space?

A vector space is a mathematical structure that consists of vectors and operations that can be performed on those vectors. These operations include addition and scalar multiplication, and they follow specific rules and properties. Vector spaces are used in linear algebra to represent geometric concepts and to solve systems of linear equations.

2. What are the basic properties of a vector space?

The basic properties of a vector space include closure under addition and scalar multiplication, associativity and commutativity of addition, existence of an additive identity element (the zero vector), existence of a multiplicative identity element (the scalar 1), and distributivity of scalar multiplication over addition. Additionally, every vector space must have a set of basis vectors that can be used to span the space.

3. How is a vector space different from a Euclidean space?

A vector space is an abstract mathematical concept that can be used to represent geometric concepts, while a Euclidean space is a specific type of vector space that follows the rules and properties of Euclidean geometry. In a Euclidean space, the vectors are represented as points in n-dimensional space, and the operations of addition and scalar multiplication are defined geometrically.

4. What is the dimension of a vector space?

The dimension of a vector space is the number of linearly independent basis vectors that can be used to span the space. This means that any vector in the space can be written as a linear combination of these basis vectors. The dimension of a vector space can be finite or infinite, depending on the number of basis vectors.

5. How is linear independence related to vector spaces?

Linear independence refers to a set of vectors that cannot be written as linear combinations of each other. In a vector space, the basis vectors should be linearly independent, meaning that no basis vector can be written as a linear combination of the other basis vectors. This property is important in determining the dimension of a vector space and in solving systems of linear equations.

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