Subspace Help: Properties & Verifying Examples

Thread starter glauss
Start date Feb 2, 2021
Tags

Linear algebra Subspaces

In summary: Neither is the case. ##\mathbf{F}## represents the field the scalars are taken from: ##b,x_2, etc.## It is stands e.g. for ##\mathbb{R}, \,\mathbb{C},\,\mathbb{Q}## or any other fields. ##\mathbf{F}^4## is a four dimensional ##\mathbf{F}-##linear vector space.In summary, the properties of subspaces are explained, and an example is given.

Feb 6, 2021

MidgetDwarf

Hmm there's also Anton. Its more applied, but proofs are gi

glauss said:

I will give that linked PDF a read, on matricies and group theory, it only looks like a few pages.
Do you have another recommendation for logic and proofs? I’d like to bring my skills up to par quickly and get back to working through Axler ASAP...

https://www.people.vcu.edu/~rhammack/BookOfProof/

Here you go.. I found this book much clearer than the Velheim book. Solutions are given to some problems in the back. I am not familiar with this edition, but it should still be the same.

<h2>1. What is subspace in mathematics?</h2><p>Subspace in mathematics refers to a subset of a vector space that satisfies all the properties of a vector space. This means that it must contain the zero vector, be closed under addition and scalar multiplication, and have a finite number of vectors.</p><h2>2. How do you verify if a set is a subspace?</h2><p>To verify if a set is a subspace, you need to check if it satisfies all the properties of a vector space. This includes checking if the set contains the zero vector, if it is closed under addition and scalar multiplication, and if it has a finite number of vectors. If all of these conditions are met, then the set is a subspace.</p><h2>3. What is the difference between a subspace and a vector space?</h2><p>A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it is also a vector space, but it is a smaller subset of the larger vector space. A vector space, on the other hand, is a set of vectors that satisfy certain properties, such as closure under addition and scalar multiplication.</p><h2>4. Can a subspace have an infinite number of vectors?</h2><p>Yes, a subspace can have an infinite number of vectors as long as it satisfies all the properties of a vector space. For example, the set of all real numbers is an infinite subspace of the vector space of all complex numbers.</p><h2>5. How are subspaces used in real-world applications?</h2><p>Subspaces are used in real-world applications to model and solve problems in various fields, such as physics, engineering, and computer science. For example, in physics, subspaces are used to represent different states of a physical system, while in computer science, subspaces are used to represent data structures and algorithms for efficient computation.</p>

1. What is subspace in mathematics?

Subspace in mathematics refers to a subset of a vector space that satisfies all the properties of a vector space. This means that it must contain the zero vector, be closed under addition and scalar multiplication, and have a finite number of vectors.

2. How do you verify if a set is a subspace?

To verify if a set is a subspace, you need to check if it satisfies all the properties of a vector space. This includes checking if the set contains the zero vector, if it is closed under addition and scalar multiplication, and if it has a finite number of vectors. If all of these conditions are met, then the set is a subspace.

3. What is the difference between a subspace and a vector space?

A subspace is a subset of a vector space that satisfies all the properties of a vector space. This means that it is also a vector space, but it is a smaller subset of the larger vector space. A vector space, on the other hand, is a set of vectors that satisfy certain properties, such as closure under addition and scalar multiplication.

4. Can a subspace have an infinite number of vectors?

Yes, a subspace can have an infinite number of vectors as long as it satisfies all the properties of a vector space. For example, the set of all real numbers is an infinite subspace of the vector space of all complex numbers.

5. How are subspaces used in real-world applications?

Subspaces are used in real-world applications to model and solve problems in various fields, such as physics, engineering, and computer science. For example, in physics, subspaces are used to represent different states of a physical system, while in computer science, subspaces are used to represent data structures and algorithms for efficient computation.