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Question
I am having difficulty understanding this question. Any help would be greatly appreciated.
Consider the matrix

A =
[2, 1, 3]
[4, 1, 4]

(a) Find the null space N(A) of A. Describe it geometrically.

(b) Find an equation for the row space R(A) of A. Describe it geometrically.
(c) Show that N(A) is perpendicular to R(A) , where N(A) stands for the null space of  A and R(A) the row space of A.

Answer
a) By definition, the null space of a matrix is the space spanned the vectors solving the homogeneous system of equations

Ax = 0.

In other words, the vectors subspace that "gets sent" to 0 (null) by the transformation A. So we need to solve

2x+y+3z = 0
4x+y+4z = 0

Since we have 2 eqns and 3 unknowns, we know one of the vector components (x, y, z) is going to be undetermined (arbitrary), but we cacn solve for the other 2; I get for z = 1 (arbitrary)

x = (-1/2)z and y = (-2)z, so the null space is given by c(-1/2,-2,1) where c = arbitrary constant. This is the eqn for a line (-1/2)x+(-2)y+z = 0 in R^3.

b) The row space of A is the space spanned by the rows of A. These are vectors, r1 and r2, which form a plane in R^3; the eqn for this plane can be found in the usual way by finding the normal vector to the plane given by r1xr2, for which I get n = <1,4,-2> and the eqn.

x+4y-2z = 0

c) To show N(A) and R(A) are orthogonal, calculate the dot (inner) product of the vector given for N(A) with r1 and r2 and show they are equal to 0 (they are), i.e., (-1/2, -2, 1)・(2,1,3) and (-1/2,-2,1)・(4,1,4).

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