# Suppose that v1 = (1,0,-2), v2 = (0,3,6), v3 =(-4,-2,3) and b=(4,1,-4) (i) Determine whether b E span (v1,v2,v3)

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### 1 Answer

You need to prove that b belongs to the span `(v_1,v_2,v_3)` , hence, you should check if there are `a,b,c` such that:

`b = a*v_1 + b*v_2 + c*v_3`

You need to convert the equation `b = a*v_1 + b*v_2 + c*v_3` into a system of linear equations, such that:

`{(4 = a*1 + b*0 + c*(-4)),(1 = a*0 + b*3 + c*(-2)),(-4 = a*(-2) + b*6 + c*3):}`

You need to solve for `a,b,c` the system of equations, such that:

`a - 4c = 4 => a = 4 + 4c`

`3b - 2c = 1 => b = (1 + 2c)/3`

`-2a + 6b + 3c = -4 => -2(4 + 4c) + 6*(1 + 2c)/3 + 3c = -4`

`-8 - 8c + 2 + 4c + 3c = -4 => -c = 2 => c = -2`

`a = 4 - 8 => a = -4`

`b = (1 - 4)/3 => b = -1`

**Hence, `b = -4v_1 - v_2 - 2v_3` belongs to the span **`(v_1,v_2,v_3).`