Determine if W = {(a,b,c)| a = b^2} is a Subspace of the Vector Space R^3

TL;DR

Determining if a set is a subspace of R cubed through specific conditions; ultimately not meeting all criteria.

Transcript

we have a set W and we're being asked if it is a subspace of R cubed so over here I've written down what it means for a set W to be a subspace of a vector space V so there's three conditions the first one says that our set W is non empty the second one says that for all vectors x and y and W the sum is also in W this is called closure under vector ... Read More

Key Insights

• 😫 Understanding the three critical conditions for a set to be a subspace.
• 😫 Testing if a set meets the criteria through examples and calculations.
• 😫 Identifying vectors in the set and proving their closure under vector addition and scalar multiplication.

Questions & Answers

Q: What are the three conditions for a set to be considered a subspace in a vector space?

The conditions are non-emptiness, closure under vector addition, and closure under scalar multiplication in a given vector space to be termed as a subspace.

Q: How can one determine if a set is a subspace of R cubed?

By checking if the set satisfies all three conditions of non-emptiness, closure under addition, and closure under scalar multiplication, one can determine if the set is a subspace of R cubed.

Q: What process can be followed to disprove that a set is a subspace?

By finding two vectors in the set whose sum is not in the set, you can disprove that the set is closed under vector addition and, consequently, not a subspace.

Q: Why is it crucial to verify all three conditions when determining if a set is a subspace?

Verifying all three conditions ensures that the set is structurally sound and complies with the fundamental principles of being a subspace in a vector space.

Summary & Key Takeaways

• Explains conditions for a set to be a subspace in a vector space.

• Illustrates how to test if a set meets the criteria for being a subspace.

• Demonstrates the process of identifying vectors and proving that their sum is not in the set.