Understanding Subspaces in R3

The video tutorial explains the concept of a set S in R3, where the third component of each vector is zero. It explores whether S is a subspace of R3 by verifying the subspace axioms: closure under vector addition and scalar multiplication, and the presence of the zero vector in S. The tutorial demonstrates that S satisfies these conditions, confirming it as a subspace of R3.

What is the defining characteristic of the set S in R3?

• All vectors have non-zero components.

• All vectors have a zero as the third component.

• All vectors have a zero as the second component.

• All vectors have a zero as the first component.

What is the main goal when analyzing the set S?

• To determine if S is a subspace of R3.

• To calculate the length of vectors in S.

• To determine if S is a vector space.

• To find the dimension of S.

What must be shown to prove that S is closed under vector addition?

• The sum of any two vectors in S is zero.

• The sum of any two vectors in S is in S.

• The sum of any two vectors in S is in R2.

• The sum of any two vectors in S is not in S.

When adding two vectors from S, what is the third component of the resulting vector?

• The sum of the first components.

• The sum of the second components.

• Zero.

• The sum of all components.

What is required to show that S is closed under scalar multiplication?

• Any scalar times a vector in S results in a vector outside S.

• Any scalar times a vector in S results in a vector in R2.

• Any scalar times a vector in S results in a zero vector.

• Any scalar times a vector in S results in a vector in S.

What happens to the third component of a vector in S when it is multiplied by a scalar?

• It becomes the product of the scalar and the third component.

• It becomes the sum of the scalar and the third component.

• It remains zero.

• It becomes the scalar value.

Why is it important to verify the presence of the zero vector in S?

• To ensure S is a vector space.

• To ensure S is closed under addition.

• To confirm S is a subspace of R3.

• To confirm S is a subset of R2.

What is the zero vector in R3?

• (0, 0, 1)

• (0, 1, 0)

• (0, 0, 0)

• (1, 0, 0)

What does the presence of the zero vector in S indicate about S?

• S is not a subspace.

• S is a subspace of R3.

• S is a subset of R2.

• S is not closed under scalar multiplication.

What is the final conclusion about the set S?

• S is not closed under vector addition.

• S is not a subspace of R3.

• S is a subspace of R2.

• S is a subspace of R3.