Linearly Dependent Sets: Answers to Your Questions

In summary, linearly dependent sets have at least one element that can be written as a linear combination of the others. This can also be used as an alternative definition of linearly dependent sets. As for determining the linear independence of a set of 2x2 matrices, invertibility can be used as an alternative method. Thank you and God bless.
  • #1
ichigo444
12
0
Do all linearly dependent sets have elements that are linear combinations of each other? Or does this apply only to some of the Linearly Dependent sets?

And as a follow up question: How do you know if a set of 2x2 matrices is linearly dependent or linearly independent?

Thank you and may God bless you. :)
 
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  • #2
The usual definition of "linearly dependent" is

"The vectors [itex]\{v_1, v_2, v_3,\cdot\cdot\cdot, v_n\}[/itex] are linearly dependent if and only if there exist scalars [itex]a_1, a_2, a_3, \cdot\cdot\cdot, a_n[/itex], not all 0, such that [itex]a_1v_1+ a_2v_2+ a_3v_3+ \cdot\cdot\cdot+ a_nv_n= 0[/itex]".

Suppose [itex]a_k[/itex] is non-zero. Then we can write [itex]a_1v_1+ \cdot\cdot\cdot+ a_{k-1}v_{k-1}+ a_{k+1}v_{k+1}+ \cdot\cdot\cdot+ a_nv_n= -a_kv_k[/itex]. Now, since [itex]a_k\ne 0[/itex], we can divide both sides of the equation by it to get [itex]v_k[/itex] written as a linear combination of the other vectors in the set.

So, yes, if a set of vectors is linearly dependent, then at least one of them can be written as a linear combination of the others.

That works the other way, also- if at least one of the vectors can be written as a linear combination of the others, then they are linearly dependent. That can be used as an alternate definition of "linearly dependent".
 
  • #3
Wow thank you good sir/ma'am. You have been answering all my question for what seems to be ages now. Thank you very much and may i know where you are from?
 
  • #4
Sir may i add? May i use invertability as an alternative way of determining the linear independence of a set? Thank you again.
 
  • #5


I am happy to provide an answer to your questions about linearly dependent sets. To answer your first question, yes, all linearly dependent sets have elements that are linear combinations of each other. This is because a set is considered linearly dependent if at least one of its elements can be written as a linear combination of the other elements. This means that there is a relationship between the elements, and one can be expressed in terms of the others.

To answer your follow-up question, determining if a set of 2x2 matrices is linearly dependent or independent requires some calculations. One method is to use the determinant of the matrix. If the determinant is equal to zero, then the set is linearly dependent. If the determinant is not equal to zero, then the set is linearly independent. Another method is to perform row operations on the matrices and check if any of them can be reduced to a row of zeros. If this is the case, then the set is linearly dependent. If none of the matrices can be reduced to a row of zeros, then the set is linearly independent.

I hope this helps answer your questions about linearly dependent sets and how to determine if a set of 2x2 matrices is linearly dependent or independent. Remember, as scientists, it is important to always question and seek answers to better understand the world around us. Best of luck in your studies and may God bless you as well.
 

Related to Linearly Dependent Sets: Answers to Your Questions

1. What does it mean for a set to be linearly dependent?

Linear dependence refers to a set of vectors in which at least one vector can be expressed as a linear combination of the other vectors in the set. This means that one vector can be written as a sum of a scalar multiple of the other vectors in the set. In other words, the vectors are not all independent of each other.

2. How can I determine if a set is linearly dependent?

To determine if a set is linearly dependent, you can use the linear dependence test. This involves setting up a system of equations with the vectors in the set and solving for the coefficients. If there exists a non-trivial solution (where at least one coefficient is not equal to zero), then the set is linearly dependent.

3. What is the difference between linearly dependent and linearly independent sets?

A linearly independent set is a set of vectors in which no vector can be expressed as a linear combination of the other vectors in the set. In other words, the vectors are all independent of each other. This is the opposite of a linearly dependent set, where at least one vector can be expressed as a linear combination of the other vectors in the set.

4. Can a set of only two vectors be linearly dependent?

Yes, a set of only two vectors can be linearly dependent. This is because one vector can be expressed as a multiple of the other vector, making them linearly dependent. However, a set of two vectors can also be linearly independent, as long as they are not scalar multiples of each other.

5. How is the concept of linear dependence used in real-life applications?

Linear dependence is commonly used in fields such as physics, engineering, and computer science. In physics, linear dependence is used to describe the relationship between forces acting on an object. In engineering, it is used to analyze structures and design systems. In computer science, linear dependence is used to solve systems of equations and optimize algorithms.

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