This is Linear Algebra, please help!
Exercise FS.M50 Robert Beezer Suppose that A is a nonsingular matrix. Extend the four conclusions of Theorem FS in this special case and discuss connections with previous results (such as Theorem NME4) Theorem NME4: Nonsingular Matrix Equivalences, Round 4 Suppose that A is a square matrix of size n. The following are equivalent. 1. A is nonsingulan. 2. A row-reduces to the identity matrix 3. The null space of A contains only the zero vector, N(A) 0) 4. The linear system LS (A, b) has a unique solution for every possible choice of b 5. The columns of A are a linearly independent set. 6. A is invertible 7. The column space of A is Cn, C(A) C Proof (in context) /knowls/theorem NME4.knowl Suppose A is an m à n matrix with Theorem FS: Four Subsets. extended echelon form N. Suppose the reduced row-echelon form of A has r nonzero rows. Then C is the submatrix of N formed from the first r rows and the first n columns and L is the submatrix of N formed from the last m columns and the last m- r rows. Then 1. The null space of A is the null space of C, N(A) N(C) 2. The row space of A is the row space of C, R (A) R (C) 3. The column space of A is the null space of L, C(A)-N(L) 4. The left null space of A is the row space of L, C(A)-R(L) Proof (in context)