Can anyone explain why (linear algebra)

In summary, the union of subspaces V and W in R^n is not always a subspace of R^n. This can be seen through examples such as the union of two lines through the origin in R^2. Even if v is in subspace V and w is in subspace W, it is not guaranteed that v+w will be in V\cupW. A specific counterexample can be given using Vid's suggestion, where V= {(x,y)| y= x}, v= (1, 1), W= {(x,y)|y= -x}, and w= (-1, 1). In this case, v+w is not in the union of V and W.
  • #1
frasifrasi
276
0
Why for the subspaces V and W in R^n , the

union of V and W is not a subspace of R^n?

Thank you.
 
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  • #2
Look at some examples to see why such as the union of two lines through the origin in R^2.
 
  • #3
If v is in subspace V and w is in subspace W, it is not, in general, true that v+ w is in V[itex]\cup[/itex]W. You can give a specific counter example by using Vid's suggestion. Let V= {(x,y)| y= x}, v= (1, 1), W= {(x,y)|y= -x}, w= (-1, 1). Is v+ w in the union of V and W?
 

Related to Can anyone explain why (linear algebra)

1. Can anyone explain why linear algebra is important in science?

Linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It is used extensively in scientific fields such as physics, engineering, computer science, and data analysis. Linear algebra enables scientists to model complex systems and solve problems that involve multiple variables and equations, making it an essential tool for understanding the world around us.

2. Why is linear algebra considered the foundation of machine learning?

Linear algebra is the backbone of machine learning algorithms and techniques. It provides the mathematical framework for representing and manipulating data in a way that allows machines to learn from it. The use of linear algebra in machine learning enables computers to perform tasks such as classification, prediction, and pattern recognition, making it a fundamental part of this rapidly growing field.

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In computer graphics and game development, linear algebra is used to create 3D graphics, manipulate images, and perform transformations on objects. It provides the tools for representing and manipulating geometric shapes, which are crucial for creating realistic and visually appealing graphics in video games, animations, and simulations. Linear algebra is also used in physics engines to simulate realistic movements and interactions between objects in games.

4. What are the real-world applications of linear algebra?

Linear algebra has numerous real-world applications in fields such as physics, engineering, economics, and data science. It is used to solve complex optimization problems, model systems, analyze data, and make predictions. For example, in economics, linear algebra is used to analyze supply and demand curves and make predictions about market trends. In physics, it is used to study quantum mechanics and describe the behavior of particles.

5. Can linear algebra be used in everyday life?

Although linear algebra may seem like a complex and abstract topic, its principles and techniques can be applied in everyday life. For instance, it can help with financial planning and budgeting by using matrices to represent income and expenses. It can also be used to optimize transportation routes or plan schedules by using linear programming. Additionally, understanding linear algebra can improve problem-solving skills and critical thinking abilities, which can be applied in various aspects of life.

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