solution system of AX=0

A

afik

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Is the solution group of the system A3X = 0
, Is equal to the solution group of the system AX = 0

If this is true you will prove it, if not give a counterexample.

thank you.
 
What have you tried on this problem?

If Ax= 0 then certainly A^3x=A^2(Ax)= A^2(0)= 0 for A any linear operator. So the solution set of A is a subset of the solution set of A^3. The question is "are there x that satisfy A^3x= 0 but not Ax= 0?"

Suppose D is the differentiation operator on the space of polynomials. What is D(ax^2+bx+c)? When is that 0? What about D^2(ax^2+ bx+ c)? When is that 0?
 
HallsofIvy said:
What have you tried on this problem?

If Ax= 0 then certainly A^3x=A^2(Ax)= A^2(0)= 0 for A any linear operator. So the solution set of A is a subset of the solution set of A^3. The question is "are there x that satisfy A^3x= 0 but not Ax= 0?"

Suppose D is the differentiation operator on the space of polynomials. What is D(ax^2+bx+c)? When is that 0? What about D^2(ax^2+ bx+ c)? When is that 0?
Click to expand...
Thank you, but I don't understand yet..
 
HallsofIvy said:
What have you tried on this problem?

If Ax= 0 then certainly A^3x=A^2(Ax)= A^2(0)= 0 for A any linear operator. So the solution set of A is a subset of the solution set of A^3. The question is "are there x that satisfy A^3x= 0 but not Ax= 0?"

Suppose D is the differentiation operator on the space of polynomials. What is D(ax^2+bx+c)? When is that 0? What about D^2(ax^2+ bx+ c)? When is that 0?
Click to expand...
Also I forgot to say that A is Matrix n by n.
 
afik said:
Is the solution group of the system A3X = 0
, Is equal to the solution group of the system AX = 0

If this is true you will prove it, if not give a counterexample.

thank you.
Click to expand...
Please post EXACT wording of the problem.

This is a homogeneous equation - so it is restricted.
 
afik said:
Is the solution group of the system A3X = 0
, Is equal to the solution group of the system AX = 0

If this is true you will prove it, if not give a counterexample.

thank you.
Click to expand...
What have you learned about solution of "homogeneous" linear equations?

Please consult your textbook/class-notes/Google and tell us what you found.
 
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