# Consider the matrix A = [[2, 1, 3],[4, 1, 4]]. (a) Find the null space N(A) of A. Describe it geometrically.

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### 1 Answer

You need to use the definition of null space of matrix, such that:

`A*N(A) = 0`

`N(A)` represents the set of vectors such that `A*N(A) = 0`

Performing the multiplication yields:

`((2,1,3),(4,1,4))*((x_1),(x_2),(x_3)) = ((0),(0))`

`((2x_1 + 1*x_2 + 3*x_3),(4*x_1 + 1*x_2 + 4*x_3)) = ((0),(0))`

You need to solve the system considering the unknown `x_3` as a free variable `alpha` , such that:

`{(2x_1 + x_2 = -3alpha),(4x_1 + x_2 = -4alpha):}`

`2x_1 + x_2 - 4x_1 - x_2 = -3alpha + 4alpha`

Reducing duplicate members, yields:

`-2x_1 = alpha => x_1 = -alpha/2`

`x_2 = -3alpha + alpha => x_2 = -2alpha`

**Hence, evaluating the null space of matrix `A` yields the collection of vectors of form `N(A) = ((-alpha/2),(-2alpha),(alpha))` that satisfy the equation **`A*N(A) = 0.`