Geometric Interpretation

Question 1

What is a degree of freedom?

Answer 1

dim(ker(A))

Question 2

Consider A*x = b:
What is the space called when b = 0?
What is the space called when b ~= 0?

Answer 2

When b = 0, this is a vector space called ker(A).

When b ~= 0, this is an affine space with dim(ker(A)).

Question 3

What is an affine space?

Answer 3

A space defined by one particular vector plus a vector space, i.e. points.

Question 4

What is the dimension of the space of solutions (vector and affine)?

Answer 4

This is dim(ker(A)).

It is the number of all-zero rows in ^A when A is a square matrix.

Question 5

What do you do when n > m?

Answer 5

Add the excess number of variables from (n - m) - this will be the dimension of the affine space.

Question 6

How can you check the dimension of the space of solutions against the rank?

Answer 6

rank(A) + dim(ker(A)) = n (number of rows).

Question 7

What is ker(A)?

Answer 7

The collection of all vectors that satisfy A*x = 0.