Linear independence
Vectors are independent if none of them can be built from the others by scaling and adding. Otherwise they are dependent.
Last updated
Analogy
Ingredients that each add something new, rather than repeating what the others already give.
Example
(1,2) and (2,4): (2,4) = 2(1,2), dependent.(1,0) and (0,1): neither is a multiple.So they are independent.Test: a(1,0) + b(0,1) = 0 forces a = b = 0.
In practice
Independence decides whether a set of vectors is an efficient description of a space.
Practice
Flashcards
- Are (1,2) and (2,4) independent?No. The second is twice the first.
Interview questions
- Explain linear independence and test whether (1,2) and (2,4) are independent.
- Independent if the only combination giving zero is trivial
- (2,4) = 2(1,2)
- So 2(1,2) - (2,4) = 0 with non-zero coefficients
- Hence dependent
Questions people ask
How do I test linear independence?
Write a combination of the vectors equal to zero and solve for the coefficients. If the only solution is all coefficients zero, the vectors are independent. If another solution exists, such as 2 times (1,2) minus (2,4) equals zero, they are dependent.
Can three vectors in the plane be independent?
No. The plane has only two dimensions, so any third vector can be built from two independent ones. Any set of more than n vectors in n-dimensional space must be dependent. This links independence to dimension. Extra vectors are always redundant, because they add no new direction to the set.