Last updated April 10, 2022

Status: Tags: #cards/math232/unit1 Links: Math 232 Course Outline

Linear Independence

Principles

?

• 
• Isolalte one, solve for other using newly found
• Otherwise, it is linearly dependent

Graphed

Checking

• For 2 vectors, check if multiple
• For sets with 3+ vectors, see if it is possible to combine 2+ to create the rest
  • Can also apply constants, set to 0, then isolate a variable and work way up to see if it is possible to find a value for all of them under one specific case

{v1, v2} are linearly dependent if and only if ;; v1 and v2 are parallel

{v1, v2, v3} are linearly dependent if and only if ;; v1, v2, v3 lie in the same plane.

Check linear independence of using matrix ?

• Each vector as a column, find solution other than 0

Check if s is linearly independent ?

• Last column shows us that m4 = m1+m2+m3

Theorems

A set S with two or more vectors in Rn is linearly dependent if and only if at least one of the vectors in S is expressible as a linear combination of the other vectors in S. ?

• Prove both ways
• 
• 
  • There must be some other constant that negates the 1

Examples

Properties

Are the following sets linearly independent?

• (a) S = {(1, 0, 0),(0, 0, 1)}
• (b) S = {(0, 1, 0),(0, 3, 0)} ?
• a) Independent since no coefficient on one point can be applied to equal the other point
• b) Dependent since 2nd point can be multipled by 1/3 to be the same as the first point

Suppose u and v are non-zero vectors. Are the following sets linearly independent?

• (a) {u}
• (b) {u, v, 0} ?
• a) Yes only if u != 0, since if it were 0 then any coefficient would lead to 0
• b) No since any number could be applied as $c_3$, making it non linearly independent
  • If there is a zero vector in a collection, it is dependent

Checking Linear Independence

Are the vectors (5, 4),(−1, 6) linearly independent? ?

• Yes
• Apply rule, isolate one variable and plug into other eqn

Are the vectors (1, 0, 1),(−1, 0, 1),(2, 0, 2) linearly independent? What do the vectors span? ?

• Not linearly independent
• Use system of elimination to isolate one and plug into others
  • 
• 
  • Set one of the variables to anything, then find out what the other variable should be
    • Variables are not all 0, but statement is still true, so not linear independent
• Span can be reduced to {(1 0 1), (-1 0 1)} since (2 0 2) is multiple of (1 0 1)
• 

Express vector a as a linear combination other vectors B and C ?

• A = t(B) + s(C) for some values t and s

Backlinks

1

list from Linear Independence and Dependence AND !outgoing(Linear Independence and Dependence)

References:

Created:: 2022-01-11 20:12