Spanning sets Find the value of a for which v 10 a 2 12 is

Find the value of a for which v = [-10 a -2 12] is in the set H = span{[5 -2 -1 -5],[0 1 3 3], [0 0 5 2]} a =

Solution

let

[-10 a -2 12] = x[5 -2 -1 -5] + y[0 1 3 3] + z[ 0 0 5 2]

=>

-10 = 5x, => x = -2

a = -2x+y => a = 4+y

-2 = -x +3y+5z => 3y+5z = -4

12 = -5x+3y+2z => 3y+2z = 2

=>

z = -2 => y = 2 => a = 6