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Determine Whether The Following Sets Are Subspaces of R3 Calculator

Reviewed by Calculator Editorial Team

This calculator helps you determine whether a given set in R³ (three-dimensional space) is a subspace. Understanding subspaces is fundamental in linear algebra, and this tool provides a step-by-step method to verify the subspace criteria.

What is a Subspace?

A subspace of a vector space is a subset that is itself a vector space under the same operations. In R³, which represents all possible three-dimensional vectors, a subspace must satisfy three key properties:

  1. Closure under addition: The sum of any two vectors in the set must also be in the set.
  2. Closure under scalar multiplication: Multiplying any vector in the set by a scalar must result in a vector that's still in the set.
  3. Contains the zero vector: The set must include the origin (0,0,0).

If a set meets all three criteria, it's a subspace of R³.

Subspace Criteria in R³

To verify if a set is a subspace, you must check the three conditions mentioned above. The calculator automates this process, but understanding the underlying principles is valuable:

Subspace Verification Steps

  1. Check if the zero vector is in the set.
  2. For any two vectors in the set, verify their sum is also in the set.
  3. For any vector in the set and any scalar, verify the scaled vector is still in the set.

If all three conditions are satisfied, the set is a subspace. If any condition fails, it's not a subspace.

How to Use This Calculator

Using the calculator is straightforward:

  1. Enter the vectors that define your set in R³.
  2. Click "Calculate" to determine if the set is a subspace.
  3. Review the result and explanation.
  4. Use the reset button to clear the inputs and start over.

The calculator provides a clear result and explains why the set is or isn't a subspace.

Examples of Subspaces and Non-Subspaces

Here are some examples to illustrate the concepts:

Example 1: A Subspace

Consider the set of all vectors where the third component is zero: {(x, y, 0) | x, y ∈ ℝ}. This set is a subspace because:

  1. The zero vector (0, 0, 0) is included.
  2. The sum of any two vectors in the set will have a zero third component.
  3. Scaling any vector in the set preserves the zero third component.

Example 2: Not a Subspace

Consider the set of all vectors where the third component is equal to the first component: {(x, y, x) | x, y ∈ ℝ}. This is not a subspace because:

  1. The zero vector (0, 0, 0) is included, but...
  2. If you take two vectors (1, 0, 1) and (0, 1, 0), their sum is (1, 1, 1), which is not in the set.

Remember: For a set to be a subspace, all three conditions must be satisfied simultaneously.

FAQ

What is the difference between a subset and a subspace?

A subset is simply a collection of elements within a larger set. A subspace is a special type of subset that is itself a vector space, satisfying the closure properties under addition and scalar multiplication.

Can a subspace be finite?

No, a subspace must be infinite. The only finite subspace is the trivial subspace containing only the zero vector.

How do I know if a set is a subspace?

You need to verify the three subspace criteria: closure under addition, closure under scalar multiplication, and containing the zero vector.

What if I'm not sure about the zero vector condition?

If the zero vector isn't in the set, it automatically fails the subspace test. The calculator checks this first.