Linear Algebra- Linear Transformations

B

Bin222

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Feb 3, 2015
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6
Can someone explain the following problem?

Let G: R3-->R3 be the function defined by the formula

G(x1,x2,x3)=(-3x1+6x2+9x3, x1-2x2-2x3, 2x1-4x2-3x3).

a)G is not onto. For which of the following triples (b1,b2,b3) is there no (x1,x2,x3) with G(x1,x2,x3)-(b1,b2,b3)?

This part I already solved by reducing the liean equations to a reduced row echelon form. I found the pivot in the 3rd row to be b1+9b2-3b3, so when it is not equal to 0 the system is inconsistent. That part was easy...However I have no idea what to do with part b and c:

b) G is not one-to-one. For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=G(0,0,0)?

A. (2,1,0)
B. (-2,1,0)
C. (2,-1,0)
D. (-2,-1,0)
E. (2,1,1)
F. none of these

b) For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=T(0,0,1)?

A. (2,1,1)
B. (2,-1,1)
C. (2,1,0)
D. (-2,1,1)
E. (-2,-1,1)
F. none of these

Thanks !!
 
Bin222 said:
Can someone explain the following problem?

Let G: R3-->R3 be the function defined by the formula

G(x1,x2,x3)=(-3x1+6x2+9x3, x1-2x2-2x3, 2x1-4x2-3x3).

a)G is not onto. For which of the following triples (b1,b2,b3) is there no (x1,x2,x3) with G(x1,x2,x3)-(b1,b2,b3)?

This part I already solved by reducing the liean equations to a reduced row echelon form. I found the pivot in the 3rd row to be b1+9b2-3b3, so when it is not equal to 0 the system is inconsistent. That part was easy...However I have no idea what to do with part b and c:

b) G is not one-to-one. For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=G(0,0,0)? Not sure if you have to show that G is not 1-1 or if you can assume it. Well what is G(0,0,0) and which of the choices below equals G(0,0,0)?

A. (2,1,0)
B. (-2,1,0)
C. (2,-1,0)
D. (-2,-1,0)
E. (2,1,1)
F. none of these

b) For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=T(0,0,1)? Don't know. What is T?

A. (2,1,1)
B. (2,-1,1)
C. (2,1,0)
D. (-2,1,1)
E. (-2,-1,1)
F. none of these

Thanks !!
Click to expand...
See comments above
 
Bin222 said:
Let G: R3-->R3 be the function defined by the formula
G(x1,x2,x3)=(-3x1+6x2+9x3, x1-2x2-2x3, 2x1-4x2-3x3).
a)G is not onto. For which of the following triples (b1,b2,b3) is there no (x1,x2,x3) with G(x1,x2,x3)-(b1,b2,b3)?

b) G is not one-to-one. For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=G(0,0,0)?
A. (2,1,0) B. (-2,1,0) C. (2,-1,0) D. (-2,-1,0) E. (2,1,1) F. none of these

b) For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=T(0,0,1)?
A. (2,1,1) B. (2,-1,1) C. (2,1,0) D. (-2,1,1) E. (-2,-1,1) F. none of these
Click to expand...

Why is unconventional notation used without explanation?
I assume that \(\displaystyle (x1,x2,x3)\) means \(\displaystyle x_1,x_2,x_3)\) then why not use \(\displaystyle (x,y,z)\) the usual?

I guess that \(\displaystyle G(x,y,z)\) is a linear transformation \(\displaystyle \mathcal{R}^3\to \mathcal{R}^3\).

Look at this to see it written correctly. That web utility can be used to help solve this. Just change the x,y,z.

Look at this the inverse of \(\displaystyle G\) exists. So is it onto?
 
pka said:
Why is unconventional notation used without explanation?
I assume that \(\displaystyle (x1,x2,x3)\) means \(\displaystyle x_1,x_2,x_3)\) then why not use \(\displaystyle (x,y,z)\) the usual?

I guess that \(\displaystyle G(x,y,z)\) is a linear transformation \(\displaystyle \mathcal{R}^3\to \mathcal{R}^3\).

Look at this to see it written correctly. That web utility can be used to help solve this. Just change the x,y,z.

Look at this the inverse of \(\displaystyle G\) exists. So is it onto?
Click to expand...



Let [FONT=MathJax_Math]G[FONT=MathJax_Main]:[/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main]3[/FONT][/FONT] be the function defined by the formula
[FONT=MathJax_Math]G​
[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]6[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main].[/FONT]​
[/FONT]​
[FONT=MathJax_Math]G[/FONT] is not onto. For which of the following triples [FONT=MathJax_Main]([FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][/FONT] is there no [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] with [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these
[FONT=MathJax_Math]G[/FONT] is not one-to-one. For which of the following triples [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] do we have [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these

For which of the following triples [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] do we have [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these

So far I have gotten a) and b), but I am still struggling with c).


 
I replaces x with x1, y with x2 and z with x3, however I am still not getting the linear equation to (0,0,1).
I chose "none of these" as an answer, but it's wrong.
I got part a and b, but I'm still struggling with part c.

This is a better display of the problem:

Let [FONT=MathJax_Math]G[FONT=MathJax_Main]:[/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]→[/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main]3[/FONT][/FONT] be the function defined by the formula
[FONT=MathJax_Math]G​
[FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]6[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]9[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main].[/FONT]​
[/FONT]​
[FONT=MathJax_Math]G[/FONT] is not onto. For which of the following triples [FONT=MathJax_Main]([FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][/FONT] is there no [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] with [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]b[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these
[FONT=MathJax_Math]G[/FONT] is not one-to-one. For which of the following triples [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] do we have [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these

For which of the following triples [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT] do we have [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]?

A. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
B. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
C. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT]
D. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
E. [FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main])[/FONT]
F. none of these
 
Bin222 said:
b) G is not one-to-one. For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=G(0,0,0)?

A. (2,1,0)
B. (-2,1,0)
C. (2,-1,0)
D. (-2,-1,0)
E. (2,1,1)
F. none of these
Click to expand...
What did you get when you plugged (0, 0, 0) into G? What did you get when you plugged each of these listed triples into G? Did any of the listed triples lead to a result which matched G(0, 0, 0)?

Bin222 said:
b) For which of the following triples (x1,x2,x3) do we have G(x1,x2,x3)=T(0,0,1)?

A. (2,1,1)
B. (2,-1,1)
C. (2,1,0)
D. (-2,1,1)
E. (-2,-1,1)
F. none of these
Click to expand...
The original question asked for this exercise still stands: What is T? ;)
 
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