jernplate8

2021-01-04

Whether the statement “ Let A be an $m\text{}\times \text{}n$ matrix.

The system$Ax=b\text{}\text{is consistent for all b in}\text{}{R}^{m}\text{}\text{if and only if the columns of A form a generating set for}\text{}{R}^{m}$ " is true or false.

The system

Liyana Mansell

Skilled2021-01-05Added 97 answers

Result used:

Consider the system$Ax=b,\text{}\text{where A is an}\text{}m\text{}\times \text{}n\text{matrix, x is a vector in}\text{}{R}^{m}.$

This system is equivalent to the system${T}_{A}(x)=b\text{}\text{and the following are the nature of solutions of}\text{}Ax=b$ and properties of

a)$Ax=b\text{}\text{has a solution if and only if b is in the range of}\text{}{T}_{A}.$

b)$Ax=b\text{}\text{has a solution for every b if and only if}\text{}{T}_{A}$ is onto.

c)$Ax=b\text{}\text{has at most one solution for every b if and only if}\text{}{T}_{A}$ 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$Ax=b\text{}\text{is consistent for all b if and only if the transfomlation}\text{}{T}_{A}$ 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.

Consider the system

This system is equivalent to the system

a)

b)

c)

Theorem used:

Let

Then, the following statements are equivalent.

a) Tis onto, that is, the range of T is

b) The columns of A form a generating set for

d) Rank

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

By the above theorem, is T is onto, then the columns of A form a generating set for

Therefore, the given statement is True.

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