Question

"(linear algebra)

how do you determine if a subspace is in R3. please provide the
algorithm and step by step expample "

Answer 1

To determine if a subspace is in R3, confirm it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication. If all three conditions are met, the subset qualifies as a subspace of R3.

Step-by-step explanation:

To determine if a subspace is in R3, you need to confirm that the subset of R3 satisfies three main properties: it contains the zero vector, it is closed under addition, and it is closed under scalar multiplication. Let's walk through an example step by step.

1. Firstly, identify the subset of R3 in question, such as a line or a plane through the origin.
2. Check if the zero vector (0, 0, 0) is in the subset. If not, it's not a subspace.
3. Take two arbitrary vectors from the subset, say v1 and v2, and verify that their sum v1 + v2 is also in the subset.
4. Take any scalar α and an arbitrary vector v from the subset and check if the product α * v is still in the subset.

If the subset you are examining passes all three conditions, it is a subspace of R3.