[Linear Algebra] - Produce an equation not implied by this system

In summary: Essentially, the Linear Combination Lemma states that any equation that can be obtained through Gaussian reduction from a given linear system is a linear combination of the equations in that system. Therefore, to produce an equation not implied by the system, you would need to come up with an equation that cannot be obtained through Gaussian reduction from the given equations.
  • #1
Tosh5457
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Homework Statement



I'm quoting the whole exercise:

The Linear Combination Lemma says which equations can be gotten from Gaussian reduction from a given linear system.

Produce an equation not implied by this system:
3x + 4y = 8
2x + y = 3

Homework Equations



Linear Combination Lemma: A linear combination of linear combinations is a linear combination.

The Attempt at a Solution



What does it mean to produce an equation not implied by the system? Is it an equation that can't be derived from the system? In that case, that will be an equation such that it doesn't result from any linear combination of the 2 equations on that system, right?
 
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  • #2
Tosh5457 said:

Homework Statement



I'm quoting the whole exercise:

The Linear Combination Lemma says which equations can be gotten from Gaussian reduction from a given linear system.

Produce an equation not implied by this system:
3x + 4y = 8
2x + y = 3

Homework Equations



Linear Combination Lemma: A linear combination of linear combinations is a linear combination.

The Attempt at a Solution



What does it mean to produce an equation not implied by the system? Is it an equation that can't be derived from the system? In that case, that will be an equation such that it doesn't result from any linear combination of the 2 equations on that system, right?

Yes, that would be my interpretation.
 

FAQ: [Linear Algebra] - Produce an equation not implied by this system

What is linear algebra?

Linear algebra is a branch of mathematics that deals with studying linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical tools to solve problems related to systems of linear equations.

How is linear algebra used in science?

Linear algebra has a wide range of applications in science, including physics, engineering, computer science, economics, and statistics. It is used to solve problems related to systems of linear equations, analyze data, and create mathematical models to represent real-world phenomena.

What is the purpose of producing an equation not implied by a system in linear algebra?

Producing an equation not implied by a system in linear algebra can help to identify the unique solutions to a system of equations. It can also be used to generate new equations that can be used to solve real-world problems or to gain a better understanding of a system.

Can you provide an example of an equation not implied by a system in linear algebra?

Yes, an example of an equation not implied by a system in linear algebra is the equation ax + by = c, where a, b, and c are constants and x and y are variables. This equation is not implied by any system of linear equations and can be used to represent a line in the x-y plane.

How does linear algebra relate to other branches of mathematics?

Linear algebra has connections to other branches of mathematics, such as calculus, differential equations, and abstract algebra. It is also used in conjunction with other mathematical tools, such as geometry and trigonometry, to solve problems in various fields of science and engineering.

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