Question: Let A be a 7 x 5 matrix with rank equal to 4 and let b be a vector in R8. The four fundamental subspaces

Let A be a 7 x 5 matrix with rank equal to 4 and let b be a vector in R8. The four fundamental subspaces associated with A are R(A), N(AT), R(AT), and N(A).
(a) What is the dimension of N(AT) and which of the other fundamental subspaces is the orthogonal complement of N(AT)?
(b) If x is a vector in R(A) and ATx = 0, then what can you conclude about the value of ||x||? Explain.
(c) What is the dimension of N(ATA)? How many solutions will the least squares system Ax = b have? Explain.

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