How to Prove a Set Is a Subspace of a Vector Space

TL;DR

A set V is a subspace of R^3 if it meets three criteria: it is non-empty, closed under vector addition, and closed under scalar multiplication. Non-emptiness is satisfied by including the zero vector. Closure under addition and scalar multiplication follows from the property that the third component equals the sum of the first two components for all vectors in V.

Transcript

so we have a set which we're calling V B is the set of all of the vectors and R cubed such that the third component is equal to the sum of the first two components and we want to prove that this is a subspace of this vector space here and all of this our field will be the set of real numbers so what do we need to show we need to show three things o... Read More

Key Insights

• 🍹 V is a subspace of R^3 if the third component of a vector in V is equal to the sum of the first two components.
• 😆 The non-emptiness condition is satisfied by showing that the zero vector is in V.
• 👍 Closure under vector addition is proven by demonstrating that the sum of two vectors from V still satisfies the condition.
• 🛀 Closure under scalar multiplication is shown by multiplying a vector from V by a scalar and showing that the condition is still met.

Questions & Answers

Q: How is V defined as a set of vectors in R^3?

V consists of vectors in R^3 where the third component is equal to the sum of the first two components.

Q: What are the three conditions to prove V is a subspace?

The conditions to prove V is a subspace are non-emptiness, closure under vector addition, and closure under scalar multiplication.

Q: Can you explain the proof for the non-emptiness condition?

The proof shows that the zero vector (0, 0, 0) is in V because the sum of its first two components equals the third component (0 + 0 = 0).

Q: How is closure under vector addition proven?

By taking two vectors from V and adding their corresponding components, it is shown that the sum of the vectors also satisfies the condition of V.

Q: How is closure under scalar multiplication demonstrated?

By multiplying a vector from V by a scalar, it is shown that the resulting vector still meets the condition of V, as the sum of its first two components is equal to the third component.

Summary & Key Takeaways

• V is defined as the set of vectors in R^3 where the third component is the sum of the first two components.

• To prove V is a subspace, three conditions need to be shown: non-emptiness, closure under vector addition, and closure under scalar multiplication.

• The proof demonstrates that V satisfies each condition, making it a subspace of R^3.