w 40 31 v1 4 4 v25 2 v35 5 If possible express w as a linea

w = (-40, 31), v1 =(-4, 4), v2=(-5, 2) v3=(-5, 5) If possible, express w as a linear combination of the vectors V1, v2 and v3. otherwise, enter DNE. For example, the answer w = 4vj + 5v2 + 6V3 would be entered 4vl + 5v2 + 6v3.

Solution

let w = av1 +bv2 +cv3

=>

-4a -5b -5c = -40

4a+2b+5c = 31

=>

-3b = -9 => b = 3

=>

4a+5c = 25

one possible solution is (a,c) = (0,5)

w = 0v1-3v2+5v3

NOTE: there are infinite possibile ways to express w as a linear combination of v1,v2,v3

w = tv1 -3v2 +(5-4t/5)v3 is a linear coombination (you can substitute any value for t)