If A is a linear operator and both xv and xcv for any real n

If A is a linear operator and both x=v and x=cv (for any real number c) are solutions to Ax=b, then what can you say about b?

Solution

The problem states that, the solutions of

A X = b ____________(1)

are

X = v ______________(2)

and

X = cv ______________(3)

substituting (2) into (1) we get,

Av=b _______________(4)

substituting (3) into (1), we get

A(cv)=b _____________(5)

or

c (Av) =b ___________(6)

but usimg (4)

c b= b

hence

b=0

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