# 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 find the null space of A, hence, you need to solve the following equation, such that:

A*X = O_(2X1)

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

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

Equating the corresponding members yields:

`{(2x_1 + x_2 + 3x_3 = 0),(4x_1 + x_2 + 4x_3 = 0):}`

You may consider `x_3` as free variable and move it to the right side, such that:

`{(2x_1 + x_2 = -3x_3),(4x_1 + x_2 = - 4x_3):}`

`4x_1 + x_2 - 2x_1 - x_2 = -4x_3 + 3x_3 => 2x_1 = -x_3 => x_1 = -x_3/2`

`2*(-x_3)/2 + x_2 = -3x_3 => x_2 = -3x_3 + x_3 => x_2 = -2x_3`

**Hence, the null space of matrix A is the collection of alll vectors of form **`N(A) = ((-x_3/2),(-2x_3),(x_3)).`