Linear Independence Question

In summary, the question is asking to determine all values of the constant k for which the given set of vectors is linearly independent in \mathbb R^4. The approach taken is to set up a coefficient matrix and try to convert it to REF. However, it may be easier to check the determinant of the matrix instead.
  • #1
Nathew

Homework Statement


Determine all values of the constant k for which the given set of vectors is linearly independent in [itex]\mathbb R^4[/itex].
{(1, 1, 0, −1), (1, k, 1, 1), (4, 1, k, 1), (−1, 1, 1, k)}

Homework Equations





The Attempt at a Solution



So far I set up a coefficient matrix
[tex]
\begin{pmatrix}
1 & 1 & 4 & -1 \\
1 & k & 1 & 1 \\
0 & 1 & k & 1 \\
-1 & 1 & 1 & k
\end{pmatrix}
[/tex]

And tried converting it to REF

[tex]
\begin{pmatrix}
1 & 1 & 4 & -1 \\
0 & 1 & k & 1 \\
0 & 0 & (-k^2+k-3) & (3-k) \\
0 & 0 & (5-2k) & (k-3)
\end{pmatrix}
[/tex]

I'm not sure if I should keep going trying to reduce this to REF to see which values of k will not work, but it just seems too messy.

Am I approaching this the wrong way?
 
Physics news on Phys.org
  • #2
Unlikely as it may seem, it looks to me like checking the determinant is as easy or easier than row reduction. If you add the last row to each of the first two rows you get a column with 3 zeros leaving you with one 3x3 determinant, which is easy to just expand.
 

What is linear independence?

Linear independence refers to a set of vectors in a vector space that cannot be written as a linear combination of each other. In other words, no vector in the set can be created by multiplying another vector by a constant and adding it to the other vectors in the set.

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

A set of vectors is considered linearly independent if and only if the only solution to the equation a1v1 + a2v2 + ... + anvn = 0 is a1 = a2 = ... = an = 0. This means that the only way to get a linear combination of the vectors to equal zero is if all of the coefficients are equal to zero.

What is the significance of linear independence?

Linear independence is important in linear algebra because it helps determine the span and basis of a vector space. It also allows for the creation of a coordinate system that is unique and consistent, which is necessary for solving systems of linear equations and other mathematical problems.

Can a set of linearly dependent vectors be linearly independent?

No, a set of vectors cannot be both linearly dependent and independent. This is because if a set is dependent, it means that one or more vectors can be written as a linear combination of the others, which violates the definition of linear independence.

How is linear independence used in real-world applications?

Linear independence is used in various fields such as physics, engineering, and computer science to solve problems involving multiple variables. It is also used in data analysis and machine learning to determine the most important features and relationships within a dataset.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
195
  • Calculus and Beyond Homework Help
Replies
3
Views
444
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
464
  • Calculus and Beyond Homework Help
Replies
2
Views
622
  • Calculus and Beyond Homework Help
Replies
6
Views
594
  • Calculus and Beyond Homework Help
Replies
8
Views
119
  • Calculus and Beyond Homework Help
Replies
2
Views
927
  • Calculus and Beyond Homework Help
Replies
1
Views
556
  • Calculus and Beyond Homework Help
Replies
1
Views
253
Back
Top