Proving R3=U+V: Solving a+x=b,2a+y=c,3a+z=d

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In summary, proving that a general point (b,c,d) in R3 is in U + V involves finding an (a,2a,3a) in U and an (x,y,z) in V which add to (b,c,d).
  • #1
franky2727
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prove R3=Udirect sum V

u={(a,2a,3a)} aER
v={(x,y,z)} x,y,z ER x+y+z=0

i solved the first bit UnV=0 but I'm having problems with the R3=U+W bit, my notes say i need to be able to "solve" a+x=b,,,2a+y=c,,,3a+z=d but what does this mean? is this simply putting the b,c,d all equal to one of the a,x,y or z?

i have done 6a=b+c+d is this adequate?
 
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  • #2
franky2727 said:
prove R3=Udirect sum V

u={(a,2a,3a)} aER
v={(x,y,z)} x,y,z ER x+y+z=0

my notes say i need to be able to "solve" a+x=b,,,2a+y=c,,,3a+z=d but what does this mean? is this simply putting the b,c,d all equal to one of the a,x,y or z?

i have done 6a=b+c+d is this adequate?

Hi franky2727! :smile:

I think they mean that, to prove that a general point (b,c,d) in R3 is in U + V, you have to find an (a,2a,3a) in U and an (x,y,z) in V which add to (b,c,d) :wink:
 
  • #3
It might be helpful to think about the geometry here. The set {a, 2a, 3a} is a line in R3, while the equation x + y + z = 0 is a plane that passes through the origin, and whose normal is the vector (1, 1, 1).

What you're doing is showing that any arbitrary vector in R3 can be written as the sum of a vector along the line plus a vector that extends from the origin out to the plane somewhere.
 
  • #4
you have to find an (a,2a,3a) in U and an (x,y,z) in V which add to (b,c,d)

well isn't that just a+x=b 2a+y=c 3a+z=d
 
  • #5
franky2727 said:
you have to find an (a,2a,3a) in U and an (x,y,z) in V which add to (b,c,d)

well isn't that just a+x=b 2a+y=c 3a+z=d

Yup! :biggrin:

… but now you know why! :wink:
 

1. How do you prove that R3 equals U+V?

To prove that R3 equals U+V, we can use the given equations a+x=b, 2a+y=c, and 3a+z=d to solve for the values of a, x, y, and z. Once we have these values, we can substitute them into the equation R3=a+x+y+z and the equation U+V=a+a+a+a to show that they are equivalent.

2. Can you explain the process of solving a+x=b, 2a+y=c, and 3a+z=d?

To solve these equations, we can use algebraic manipulation and substitution. First, we can isolate the variable a in each equation by subtracting the other terms from both sides. Then, we can substitute the value of a into the other equations to solve for the remaining variables. Once we have values for all variables, we can check if they satisfy all three equations.

3. What is the importance of proving R3 equals U+V?

Proving that R3 equals U+V helps to confirm the validity of the given equations and the solution. It also demonstrates the properties of addition and substitution in algebra, which are fundamental concepts in mathematics.

4. What if the equations given do not have a solution?

If the equations do not have a solution, it means that the given statements are inconsistent and cannot be satisfied simultaneously. This could be due to an error in the equations or an incorrect assumption. In this case, the equations cannot be used to prove R3 equals U+V.

5. Are there any other methods to prove R3 equals U+V?

Yes, there are other methods to prove R3 equals U+V, such as using graphical representation or using matrices. However, the given equations a+x=b, 2a+y=c, and 3a+z=d are best solved using algebraic manipulation and substitution.

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