Linear Dependence/Independence Help

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In summary: If you are asking about how to find two specific vectors in a space, then the answer would be difficult for me to provide without more information. In summary, I am not sure where to start with this question.
  • #1
vg19
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Hi,

I am struggling with the following question and don't know where to start. I would appreciate any help.

Thanks

Determine whether the following sets of vectors form bases for two dimensional space. If a set forms a basis, determine the coordinates of v = (8,7) relative to this base

a) v1 = (1,2) v2 = (3,5)
b) v1 = (3,5), v2 = (6,10)

Thanks!
 
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  • #2
What exactly have you tried so far and what is your thinking?
 
  • #3
Hi,

To be honest, I havnt done anything with this question because I really do not know where to start. Actually, I do not really understand what this question is asking me to do in the first place. What exactly does "form bases for two-dimensional space" mean? And when they said "relative to this base" what does that mean?

Thanks
 
  • #4
I'm not sure where to begin.

In ordinary "Cartesian" space you can express any vector as a sum of the two basis vectors (1, 0) and (0, 1), i.e. the unit vectors along the x and y axes respectively. Those vectors happen to be orthogonal and, for most purposes, are quite handy and useful. However, they may not be the most convenient basis vectors in certain situations such as in analyzing certain crystal lattices where you may want basis vectors more closely resembling the structure of the lattice.

In order to accomplish that you have one overriding constraint: the basis vectors you choose must not be collinear. Your example (b) uses (3, 5) and (6, 10) which are obviously collinear since multiplying the first by 2 gives the second. Your first example with (1, 2) and (3, 5) works fine! Any vector in your 2-D space can be written as a linear combination of those two basis vectors.

All you need to do is to find the value of, say, A and B such that A*(1, 2) + B*(3, 5) gives the vector in question. In your case, find A and B such that A*(1, 2) + B*(3, 5) = (8, 7).

Didn't they cover any of this in the course you enrolled in?
 
  • #5
If you are trying to prove that a given set of vectors is a basis, then a reasonable START would be to look up the definition of "basis"!
 

1. What is linear dependence/independence?

Linear dependence and independence refer to the relationship between vectors in a vector space. If one vector can be expressed as a linear combination of other vectors, it is considered linearly dependent. If no vector can be expressed as a linear combination of other vectors, they are considered linearly independent.

2. How do you test for linear dependence/independence?

To test for linear dependence or independence, you can use the following methods:

  • Row reduction: Perform row reduction on the vectors. If a row of all zeroes is present, the vectors are linearly dependent.
  • Rank: Calculate the rank of the matrix formed by the vectors. If the rank is less than the number of vectors, they are linearly dependent.
  • Determinant: Calculate the determinant of the matrix formed by the vectors. If the determinant is zero, the vectors are linearly dependent.

3. Why is linear dependence/independence important?

Linear dependence/independence is important because it allows us to understand the relationships between vectors in a vector space. It also helps us determine if a set of vectors is a basis (linearly independent) or not.

4. Can a set of vectors be both linearly dependent and independent?

No, a set of vectors can only be either linearly dependent or independent. If a set of vectors is linearly dependent, it means that at least one vector can be expressed as a linear combination of the other vectors, making them not independent.

5. How does linear dependence/independence relate to linear transformations?

Linear dependence/independence is closely related to linear transformations. If a set of vectors is linearly independent, it forms a basis for the vector space and can be used to represent any vector in that space. This is important in linear transformations, as it allows us to understand how a transformation affects different vectors in a vector space.

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