# Linear algebra (impact of solution of theory of linear equations)

• hoju
In summary, the conversation is about a system of equations and finding values of t for which solutions can be obtained. After row-reducing the augmented matrix, it is found that if t = -21, the system has no solution. For other values of t, a solution can be obtained. The implications of this example in the theory of linear equations are discussed, with a focus on the geometric significance of the solution set. The conversation also includes the steps taken to find t and a discussion on the theory of linear equations.
hoju
I am currently taking the first college level linear algebra course and have a question.

Consider the system of equations:

1 3 2 |1
0 -4 5 |-23
2 2 9 |t

Find values of t for which solutions to this augmented matrix can be obtained. Explain the implications of this example in the theory of linear equations.

Answer:

I found that t=-21 and that the solution is of the form {x, y, z} = {-1/4(23z+65), 1/4(5z+23), z}. That much is correct. What I don't know is what the bold statement above is asking of me. Any help is appreciated. Thanks.

In row-reducing this augmented matrix, I found that if t = -21, the system has no solution. For a different value of t, I found the solution you show.

For implications of this example in the theory of linear equations, I'm not sure which implications the text author has in mind, but I would start with the geometric significance of your solution set.

Thanks for the reply Mark44. Here is the work I did to find t.

1 3 2 |1
0 -4 5 |-23
2 2 9 |t

row reducing

1 3 2 |1
0 -4 5 |-23
0 -4 5 |t-2

1 3 2 |1
0 -4 5 |-23
0 0 0 |t-2+23

1 3 2 |1
0 -4 5 |-23
0 0 0 |t+21

It follows that t=-21 is necessary for the system to have a solution so that 0=0 in the last row. Did I miss something?

I know that the system has an infinite number of solutions that depend on the value of z, but I am still not sure how this impacts the theory of linear equations. Probably because I am somewhat unfamiliar with the theory itself. The book isn't much help.

I am guessing that the theory of linear equations states that the roots of each equation in the system are the same if the system has a solution, but in this case the existence and identity of the solution are restricted by t and z, respectively. Would that be on the right track? Thanks for the help.

## 1. What is linear algebra?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of linear transformations and their properties, as well as systems of linear equations and their solutions.

## 2. What is the importance of linear algebra in science and technology?

Linear algebra is a fundamental tool in many areas of science and technology, including physics, engineering, computer science, and economics. It provides a powerful framework for modeling and solving problems involving linear relationships, making it essential for data analysis, optimization, and machine learning algorithms.

## 3. What is the impact of linear algebra on the solution of linear equations?

The theory of linear equations, which is a central topic in linear algebra, has numerous practical applications. It allows us to solve systems of equations efficiently, which is crucial for solving real-world problems in fields such as physics, engineering, and economics. Additionally, the concepts of linear independence and basis vectors play a significant role in the study of vector spaces and their transformations.

## 4. How does linear algebra relate to other branches of mathematics?

Linear algebra has connections to many other branches of mathematics, including calculus, geometry, and abstract algebra. For example, the vector calculus used in physics and engineering relies heavily on the concepts and techniques of linear algebra. In geometry, linear transformations and their properties are used to study geometric transformations such as rotations and reflections. In abstract algebra, the study of vector spaces and their operations is a fundamental part of the theory of groups and fields.

## 5. What are some real-world applications of linear algebra?

Linear algebra has numerous applications in the real world, such as image and signal processing, data compression, and computer graphics. It is also essential for solving optimization problems, which arise in fields such as logistics, finance, and operations research. Additionally, linear algebra is used extensively in machine learning and artificial intelligence, where it plays a crucial role in building and training models for tasks such as image recognition and natural language processing.

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