Linearly Dependent Sets: Answers to Your Questions

[ichigo444]

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. :)

 

[HallsofIvy]

The usual definition of "linearly dependent" is

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

Suppose a_k is non-zero. Then we can write 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. Now, since a_k\ne 0, we can divide both sides of the equation by it to get v_k 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".

 

[ichigo444]

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?

 

[ichigo444]

Sir may i add? May i use invertability as an alternative way of determining the linear independence of a set? Thank you again.