Solving for Vector Space V: Find Dimension & Basis

In summary, the conversation discusses finding the dimension and basis of a vector space V, represented by a set of vectors in R^3. The solution is found by expressing a general vector in V as a linear combination of two vectors, and determining if these two vectors are linearly independent. If so, then these two vectors form a basis of V. The conversation also clarifies that a basis of a vector space is not unique, but the chosen basis must span the vector space and be linearly independent.
  • #1
mathmathmad
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Homework Statement


Find the dimnesion and a basis of vector space V


Homework Equations


V is the set of all vectors (a,b,c) in R^3 with a+2b-4c=0


The Attempt at a Solution


(4c-2b,b,c) = b(-2,1,0) + c(4,0,1)
so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right?

how do I get the 3rd linearly independent vector so that it forms the basis of V?
Is it (1,0,0)? (0,1,0) seems possible... but the basis of a vector space isn't unique right?
 
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  • #2
mathmathmad said:
(4c-2b,b,c) = b(-2,1,0) + c(4,0,1)
This is a good start, and the right idea needed to solve the problem.

so {(-2,1,0),(4,0,1)} is the basis of the SUBSPACE of V right?
THE subspace? V has many subspaces and you're not asked to find bases of them. What you want is a basis of V. Remember that a basis of V is a set of vectors that span V and is linearly independent. Now consider:
1. For any vector of the form (4c-2b,b,c) (i.e. a vector in V) can you express it as a linear combination of (-2,1,0) and (4,0,1). If the answer is yes, then (-2,1,0) and (4,0,1) span V and therefore we need not find more basis elements.
2. Is {(-2,1,0),(4,0,1)} linearly independent?
If you can answer both 1 and 2 in the affirmative, then {(-2,1,0),(4,0,1)} is a basis of V.

how do I get the 3rd linearly independent vector so that it forms the basis of V?
Why do you believe you need a 3rd vector? You would only need a third vector if the dimension of V is 3, but it could be smaller (2 for instance).

Is it (1,0,0)? (0,1,0) seems possible... but the basis of a vector space isn't unique right?

Neither (1,0,0) nor (0,1,0) is actually in V so they can't be used. No a basis isn't unique, but your set from the first part of your answer is a good guess for a basis.
 
  • #3
I think I've read somewhere that if there is m linearly independent vectors in R^n where m<n, these vectors form the subspace

since we have R^3 now and found 2 linearly independent vectors, so I thought these 2 vectors form a subspace...

does that mean basis of V is JUST {(-2,1,0) , (4,0,1)} in this case?
 
  • #4
mathmathmad said:
I think I've read somewhere that if there is m linearly independent vectors in R^n where m<n, these vectors form the subspace
I guess I was thrown of by your use of the. It's true that m linearly independent vectors span a subspace of R^n. In your case V is the subspace of R^n.

since we have R^3 now and found 2 linearly independent vectors, so I thought these 2 vectors form a subspace...
They do, but they do form the subspace V or another?

does that mean basis of V is JUST {(-2,1,0) , (4,0,1)} in this case?
Yes, but you need to confirm it by proving that they span V and that they are linearly independent. Of course this implies that V has dimension 2. So in this case {(-2,1,0), (4,0,1)} span a subspace of R^3, and this subspace is V.
 

1. What is a vector space?

A vector space is a mathematical concept that represents a set of vectors that can be added together and multiplied by scalars. It is a fundamental concept in linear algebra and is used to describe and solve many problems in physics, engineering, and other fields.

2. How do you find the dimension of a vector space?

The dimension of a vector space is the minimum number of independent vectors that can span the entire space. To find the dimension, you can use the Rank-Nullity Theorem, which states that the dimension is equal to the rank of the matrix representing the vectors minus the nullity (dimension of the null space).

3. What is a basis of a vector space?

A basis of a vector space is a set of linearly independent vectors that can be used to represent any vector in the space. This means that any vector in the space can be written as a unique linear combination of the basis vectors. The number of basis vectors is equal to the dimension of the vector space.

4. How do you find the basis of a vector space?

To find the basis of a vector space, you can use the Row-Echelon form or Reduced Row-Echelon form of the matrix representing the vectors in the space. The non-zero rows of the reduced matrix will form the basis of the vector space.

5. Can a vector space have more than one basis?

Yes, a vector space can have more than one basis as long as the basis vectors are linearly independent and can span the entire space. This means that there can be infinitely many different bases for the same vector space, but they will all have the same dimension.

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