Question

IV. VNICI Previous Problem Problem List Next Problem (1 point) Each graph below is the graph of a system of three linear equations in three unknowns of the form Ar = b. Determine the dimension of the null space of the matrix A. dimension ( dimensioni dimension dimension dimension (Click on a graph to enlarge it.) Note: In order to get credit for this problem all answers

Answer 1

Let us first analyse what we can from the graphs. A linear equation in three variables, when plotted, will give us a plane. So a system of three linear equations in three variables will give us three planes. Now, when we plot these planes, their intersection shall give us the solution to the system. Now, three planes may or may not intersect, and even if they do, that intersection may not be unique. So, we have:

A. Two planes intersecting at a line. B. Three planes intersecting at a line. C. Three planes intersecting at a point. D. One plane. E. Three planes intersecting at a point.

Now, if you view the system in the form Ax=b, where x and b are column vectors and A is a matrix, then it is easy to view A as a linear transformation, which transforms the vector x to b. Then, solving a system of equations becomes as simple as asking, which vector, when transsformed by A, will give b. And we see that for C and E, we get a unique point as our x, i.e. A transforms ONLY x to b, i.e. for every b, we have a unique x such that Ax=b, which means that the rank of the transformation A is 3, and by the rank-nullity theorem, the nullity of A becomes 0. (You can also think of this as, A does not collapse any dimension of x and thus, it's nullity is 0). Hence, nullity of C and E = 0.

Now, in B we see that our solution set is a line i.e. a whole line, infinite no of points, get transformed to a single point b. Which means that, simply, if b was the null vector, then the situation would be the same, so we have a line (dimension 1) being nullified, i.e. the nullity, or dimension of null space of the transformation A is 1. This can also be seen by the fact that the three planes are collinear, i.e. one of them can be respresented as the linear combination of the rest=> the rank of A =2, thus, by rank-nullity theorem, the nullity would be 1.

Now, for D, we see that all three planes are the same, i.e. the solution set is a plane=> all three rows of A are multiples of each other => rank of A=1=> nullity=2.

Now, returning to A: as we cannot see the third plane, it is difficult to arrive at a conclusion. The solution does not exist. Nullity can be 0 or 1 or 2 or 3.