# Let A = [[-8,-2,-2,-4],[4,1,1,2]] Find a basis of nullspace(A).

Asked on by bobobooo

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the solution to equation `A*X = 0` and then you need to decompose the result.

`((-8,-2,-2,-4),(4,1,1,2))*((x_1),(x_2),(x_3),(x_4)) = ((0),(0))`

`((-8x_1 - 2x_2 - 2x_3 - 4x_4),(4x_1 + x_2 + x_3 + 2x_4)) = ((0),(0))`

`{(-8x_1 - 2x_2 - 2x_3 - 4x_4 = 0),(4x_1 + x_2 + x_3 + 2x_4) = 0):}`

`delta = [(-8,-2),(4,1)] ` => `delta = 0`

Since delta = 0, hence, you may consider `x_2,x_3,x_4` as free variables, such that:

`-8x_1 = 2x_2 + 2x_3 + 4x_4 => x_1 = -(x_2)/4 - (x_3)/4 - (x_4)/2`

Hence, evaluating the solution to equation `A*X = 0` yields:

`X = (( -(x_2)/4 - (x_3)/4 - (x_4)/2),(x_2),(x_3),(x_4))`

Decomposing the solution, yields:

`X = x_2*((-1/4),(1),(0),(0)) + x_3*((-1/4),(0),(1),(0)) + x_4*((-1/2),(0),(0),(1))`

Hence, evaluating the basis of null space of A yields the vectors `bar u = ((-1/4),(1),(0),(0)), bar v = ((-1/4),(0),(1),(0)), bar w = ((-1/2),(0),(0),(1))` .

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