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178              Chapter 4                                              Vector Spaces






                                                                Summary

                                       The four fundamental subspaces associated with A m×n are as follows.
                                                                                           m
                                       •   The range or column space:      R (A)= {Ax}⊆  .
                                                                                T        T     n
                                       •   The row space or left-hand range:  R A  = A y ⊆  .
                                                                                                   n
                                       •   The nullspace:                  N (A)= {x | Ax = 0}⊆  .
                                                                                T         T           m
                                       •   The left-hand nullspace:        N A    = y | A y = 0 ⊆  .
                                       Let P be a nonsingular matrix such that PA = U, where U is in row
                                       echelon form, and suppose rank (A)= r.

                                       •   Spanning set for R (A) = the basic columns in A.
                                                               T
                                       •   Spanning set for R A  = the nonzero rows in U.
                                       •   Spanning set for N (A) =the h i ’s in the general solution of Ax = 0.
                                                               T
                                       •   Spanning set for N A  = the last m − r rows of P.
                                       If A and B have the same shape, then
                                             row                              T        T
                                       •   A ∼ B ⇐⇒ N (A)= N (B) ⇐⇒ R A         = R B    .
                                             col                             T        T
                                       •   A ∼ B ⇐⇒ R (A)= R (B) ⇐⇒ N A        = N B    .



                   Exercises for section 4.2




                                    4.2.1. Determine spanning sets for each of the four fundamental subspaces
                                           associated with
                                                                                    
                                                                    1    2  1  1    5
                                                            A =    −2  −4  0  4 −2    .
                                                                    1    2  2  4    9



                                    4.2.2. Consider a linear system of equations A m×n x = b.
                                              (a) Explain why Ax = b is consistent if and only if b ∈ R (A).
                                              (b) Explain why a consistent system Ax = b has a unique solution
                                                  if and only if N (A)= {0}.
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