What Values of k Make These Vectors Linearly Dependent?

In summary, we have 4 vectors with different combinations of 1s and k. We need to determine for which values of k the vectors will be dependent. Using the Gaussian method on a matrix representation of the vectors, we get a reduced matrix with a solution of (0,0,0,0), indicating that the vectors are independent. However, upon further examination, we see that when k=1 or -3, the system has an infinite number of solutions, making the vectors dependent. This is because when k=1, all the vectors are equal, and when k=-3, the sum of the vectors is the null vector.
  • #1
Yankel
395
0
I have 4 vectors:

(1,1,1,k), (1,1,k,1), (1,k,1,1), (k,1,1,1)

I need to find out for which values of k (if any) the vectors will be dependent.

I put them all to a matrix, and reduced it using the Gaussian method (homogeneous system). I received:

1st row: 1 1 1 k 0

2nd row: 0 (k-1) 0 (1-k) 0

3rd: 0 0 (k-1) (1-k) 0

4th: 0 0 0 (-2k+3-x^2) 0the reduction is correct, I verified with MAPLE.

I am stuck here, MAPLE say the solution is (0,0,0,0), meaning that the vectors are independent, but on the other hand, when I do it manually, I think is has a solution, for example, the last equation is:

(-2k+3-x^2)*x4=0

it can be either when x4 is 0 or when k is 1 or -3. I am confused...(Worried) :confused:
 
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  • #2
What are x and x4?

When k=1 all the vectors are equal, so that's a solution.
When k=-3 the sum of the vectors is the null vector, so that's also a solution.
 
Last edited:
  • #3
ah, sorry, confused x with k.

when k=1,-3 the system has infinite number of solutions, right ? so for these values the vectors are dependent...got it now, thanks !
 

FAQ: What Values of k Make These Vectors Linearly Dependent?

What are linear dependent vectors?

Linear dependent vectors are a set of vectors that can be expressed as a linear combination of other vectors in the same space. This means that one or more vectors in the set can be written as a scalar multiple of another vector in the set.

How do you determine if a set of vectors is linearly dependent?

A set of vectors is linearly dependent if there exists at least one vector in the set that can be written as a linear combination of the other vectors in the set. This can be determined by setting up a system of equations and solving for the coefficients.

Why is it important to understand linear dependent vectors?

Understanding linear dependent vectors is important because they provide insight into the structure and relationships within a set of vectors. It also allows for simplification and optimization of calculations involving these vectors.

Can linearly dependent vectors span the entire vector space?

Yes, linearly dependent vectors can span the entire vector space. This means that even though they are not all linearly independent, they can still represent all possible vectors in that space through linear combinations.

How can linearly dependent vectors be used in practical applications?

Linearly dependent vectors can be used in a variety of practical applications, such as in data analysis and machine learning. They can also be used in physics and engineering to model and analyze complex systems.

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