# 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.

Whether the statement “ Let A be an matrix.
The system " is true or false.
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Liyana Mansell
Result used:
Consider the system
This system is equivalent to the system and properties of
a)
b) is onto.
c) is one-to-one.
Theorem used:
Let $T:{R}^{m}\to {R}^{m}$ be a linear transformation with standard matrix A.
Then, the following statements are equivalent.
a) Tis onto, that is, the range of T is ${R}^{m}.$
b) The columns of A form a generating set for ${R}^{m}.$
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 is onto.
By the above theorem, is T is onto, then the columns of A form a generating set for ${R}^{m}.$
Therefore, the given statement is True.