Let a1 For what values of h is b in the plane spanned by a1

Let a_1 = For what value(s) of h is b in the plane spanned by a_1 and a_2?

Solution

The vector \"b\" is in the span of a₁ and a₂ if there is a solution to the linear system of equation

a₁ x₁ + a₂ x₂ = b

or

1                                         -6                                        5

4       x₁             +               -20           x₂           =           4

-1                                          2                                         h

the augmented matrix is

1                -6                5

4               -20               4

-1                 2                h

we have to reduce it in row echelon form

R₃ -> R₃  + R₁

1                -6                5

4               -20               4

-1+1          2+(-6)          h+5           // +(-a)   = -a

or

1                -6                5

4               -20               4

0                -4                h+5

R₂ -> R₂ -4R₁

1                -6                   5

4-4          -20-4(-6)          4-4(5)

0                -4                h+5

or

1                -6                5

0                4                -16

0                -4               h+5

R₃ -> R₃ +R₂

1                -6                5

0                4                -16

0             -4 +4           h+5+(-16)

or

1                -6                5

0                4                -16

0                0                h-11

if h=11 in the 3rd row then there will be many solutions.

therefore b in the plane spanned by a1 and a2 only when h=11