The Equation Ax 0 has a unique solution Let A be an m n ma

The Equation Ax = 0 has a unique solution.

Let A be an m × n matrix. Consider the following statement: Based on the given statement (i.e., the boxed one above), which of the following statements can you deduce? In other words, which of the following statements are implied by the given statement?

(a) All columns of A are pivot columns.

(b) The equation Ax = b has a unique solution for every b in Rm.

(c) The columns of A are linearly independent.

(d) The columns of A span Rm.

(e) For all b in Rm such that Ax = b is consistent, the equation Ax = b has a unique solution.

(f) There is a leading entry in every row of any echelon form of A.

(g) The linear transformation T (x) = Ax is one-to-one.

(h) The linear transformation T (x) = Ax is onto.

Please help and try to explain why the choices are correct. Thanks :)

Solution

(a) All columns of A are pivot columns.

False: If A is mxn and the equation Ax = b is consistent for every b in R^m, then A must have m pivot columns.If A has m pivot columns, then the reduced matrix cannot have a row of zeroes.

(b) The equation Ax = b has a unique solution for every b in Rm.

False: Either a system has no solutions (inconsistent), one solution (consistent with no free variables means unique solution) or an infinite number of solutions (consistent with one or more free variables).

(c) The columns of A are linearly independent.

False: The columns of matrix A are linearly independent if and only if the equation Ax=0 has only the trivial solution

(d) The columns of A span Rm.

The columns of A span Rm.

• The equation Ax= b is consistent for every b in Rm.

• The equation Ax=b has at least one solution for every b in Rm.

• The equation T(x) =b has at least one solution for every b in Rm.

(e) For all b in Rm such that Ax = b is consistent, the equation Ax = b has a unique solution.

False: The system Ax=b is inconsistent when we obtain a zero row in the reduced row echelon form of A which corresponds to a nonzero entry in the correspondingly reduced form of b. There are never nonzero entries in the reduced form of 0, and so Ax= 0 is always consistent.

(f) There is a leading entry in every row of any echelon form of A.

A leading entry of a row is the leftmost nonzero entry in a nonzero row.

A rectangular matrix is in echelon form if it has the following properties:

(1) All nonzero rows are above any rows of all zeros.

(2) Each leading entry is in a column to the right of the leading entry of the row above it.

(3) All entries in a column below the leading entry are all zeros.

g) The linear transformation T (x) = Ax is one-to-one.

T is not one-to-one. Because Ax = 0 has infinitely many solutions.

T is one-to-one if and only if the equation Ax = 0 has only the trivial solution.

(h) The linear transformation T (x) = Ax is onto.

Yes, T is onto. Because, the number of pivot positions is equal to the number of rows