Whether the statement “ Let A be an m × n matrix. The system Ax=b is consistent for all b in Rm if and only if the columns of A form a generating set for Rm " is true or false.

Question

Whether the statement “ Let A be an m × n matrix.   The system Ax=b is consistent for all b in Rm if and only if the columns of A form a generating set for Rm &quot; is true or false.

Liyana Mansell · Accepted Answer

Result used:  Consider the system Ax=b, where A is an m × nmatrix, x is a vector in Rm.   This system is equivalent to the system TA(x)=b and the following are the nature of solutions of Ax=b and properties of  a) Ax=b has a solution if and only if b is in the range of TA.  b) Ax=b has a solution for every b if and only if TA is onto.  c) Ax=b has at most one solution for every b if and only if TA is one-to-one.  Theorem used:  Let T:Rm→Rm be a linear transformation with standard matrix A.  Then, the following statements are equivalent.  a) Tis onto, that is, the range of T is Rm.  b) The columns of A form a generating set for Rm.  d) Rank A=m  Calculation:  A system of equations is said to be consistent is there is at least one set of values of the variables, that satisfies the equations. For that, there should not be zero row in the reduced row echelon.  Note that, for a linear transformation to be onto, the number of non-zero rows in the reduced row echelon form, that is, the rank should be equal to the number of rows.  That is, the reduced row echelon form should not contain a zero row.  Then, the system Ax=b is consistent for all b if and only if the transfomlation TA is onto.  By the above theorem, is T is onto, then the columns of A form a generating set for Rm.  Therefore, the given statement is True.

Whether the statement “ Let A be an m times n matrix. The system Ax = b text{is consistent for all b in} R^{m} text{if and only if the columns of A form a generating set for} R^{m} " is true or false.

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