Proving R is Infinite-Dimensional & Solving 3 Variable System

In summary, the space R over the field of rational numbers Q with the usual operations is infinite dimensional. For the system given, values of lambda and μ can be found to determine whether the system has no solution, a unique solution, or an infinite number of solutions. If f(\mu,\lambda) is not 0, then there is a unique solution, if f(\mu,\lambda) is 0 but g(\mu,\lambda) is not 0, there is no solution, and if both f(\mu,\lambda) and g(\mu,\lambda) are 0, then there is an infinite number of solutions.
  • #1
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(1) Show that the space R over the field of rational numbers Q with the usual operations is infinite dimensional.

(2) Find the values of lambda and μ so that the system
2x1 + 3x2 + 5x3 = 9
7x1 + 3x2 − 2x3 = 8
2x1 + 3x2 + lamdax3 = μ
has (i) no solution, (ii) a unique solution, and (iii) an infinite number of
solutions.

Hints:::(2) (i) lamda= 5, μ [tex]\neq[/tex] 9 (ii) lamda [tex]\neq[/tex] 5, μ arbitrary (iii) lamda = 5, μ = 9.
 
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  • #2
1) If it the space were finite dimensional, there would exist a finite basis: a finite number of real numbers such that every real number can be written as a sum of rational numbers times those real numbers. Since the set of rational numbers is countable, that would imply that the set of real numbers is countable.

2) Row reduce the augmented matrix to a triangular form (you don't need to get 0s above the diagonal). The last row will be all 0s except for functions of [itex]\mu[/itex] and [itex]\lambda[/itex] in the last two places, corresponding to the equation [itex]f(\mu,\lambda)x_3= g(\mu,\lambda)[/itex]. If [itex]f(\mu,\lambda)[/itex] is not 0, then you can divide both sides by [itex]f(\mu,\lambda)[/itex] to get a single value for [itex]x_3[/itex]. You will then have to look at the row above to see if the coefficient of [itex]x_2[/itex] is also non-zero. Values of [itex]\mu[/itex] and [itex]\lambda[/itex] that make both of those coefficients non-zero give a unique solution.

If [itex]f(\mu,\lambda)= 0[/itex], then if [itex]g(\mu,\lambda)[/itex] not 0, there is no solution. Values of [itex]\mu[/itex] and [itex]\lambda[/itex] that make f= 0, but g not equal to 0 give no solution.

If [itex]f(\mu,\lambda)[/itex] and [itex]g(\mu,\lambda)[/itex] are both 0, you again need to look at the second row. Values of [itex]\mu[/itex] and [itex]\lambda[/itex] that make the entire row 0 give an infinite number of solutions, values that make the coefficient of \(\displaystyle x_2\) 0 while the last number in the row non-zero give no solutions.
 

1. How do you prove that R is infinite-dimensional?

To prove that R (the set of real numbers) is infinite-dimensional, we can use the fact that R is uncountable. This means that there is no one-to-one correspondence between the set of real numbers and the set of natural numbers. By contradiction, we can show that any finite-dimensional vector space must have a finite basis, which contradicts the uncountability of R. Therefore, R must be infinite-dimensional.

2. What is a 3 variable system?

A 3 variable system is a system of equations with 3 unknown variables. It can be represented in the form of Ax + By + Cz = D, where A, B, and C are coefficients and x, y, and z are the unknown variables. Solving a 3 variable system involves finding the values of x, y, and z that satisfy all of the equations in the system.

3. How do you solve a 3 variable system?

To solve a 3 variable system, we can use techniques such as substitution, elimination, or Gaussian elimination. These methods involve manipulating the equations in the system to isolate one variable and then using its value to solve for the other variables. It is also important to check the solution by plugging the values back into the original equations to ensure they all hold true.

4. Can a 3 variable system have more than one solution?

Yes, a 3 variable system can have more than one solution. In fact, there are three possible outcomes for a 3 variable system: it can have one unique solution, multiple solutions, or no solution at all. The number of solutions depends on the equations in the system and their relationship to each other.

5. Why is solving a 3 variable system important in mathematics and science?

Solving a 3 variable system is important in mathematics and science as it allows us to find the values of unknown variables in a system of equations. This can be applied in various real-world problems, such as optimizing resources in economics, analyzing chemical reactions in chemistry, and modeling physical systems in physics. Additionally, it helps us understand the relationships between different variables and how they affect each other.

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