Table of Contents

## How do you prove it is subspace?

In other words, to test if a set is a subspace of a Vector Space, you only need to check if it closed under addition and scalar multiplication. Easy! ex. Test whether or not the plane 2x + 4y + 3z = 0 is a subspace of R3.

**How do you determine if W is a subspace of R3?**

- You have to show that the set is non-empty , thus containing the zero vector (0,0,0).
- You have to show that the set is closed under vector addition.
- Lastly, the set has to be closed under scalar multiplication.
- These three things satisfied tells you if you have a subspace of R3.

### What is a subspace of R?

Any subset of R n that satisfies these two properties—with the usual operations of addition and scalar multiplication—is called a subspace of Rn or a Euclidean vector space. The set V = {(x, 3 x): x ∈ R} is a Euclidean vector space, a subspace of R2.

**What is a subspace of R2?**

Take any line W that passes through the origin in R2. If you add two vectors in that line, you get another, and if multiply any vector in that line by a scalar, then the result is also in that line. Thus, every line through the origin is a subspace of the plane.

## How do you show a vector space?

Prove Vector Space Properties Using Vector Space Axioms

- Using the axiom of a vector space, prove the following properties.
- (a) If u+v=u+w, then v=w.
- (b) If v+u=w+u, then v=w.
- (c) The zero vector 0 is unique.
- (d) For each v∈V, the additive inverse −v is unique.
- (e) 0v=0 for every v∈V, where 0∈R is the zero scalar.

**Which of the following is a subspace of R 3?**

If you did not yet know that subspaces of R3 include: the origin (0-dimensional), all lines passing through the origin (1-dimensional), all planes passing through the origin (2-dimensional), and the space itself (3-dimensional), you can still verify that (a) and (c) are subspaces using the Subspace Test.

### Is R2 a subspace of R?

Instead, most things we want to study actually turn out to be a subspace of something we already know to be a vector space. However, R2 is not a subspace of R3, since the elements of R2 have exactly two entries, while the elements of R3 have exactly three entries. That is to say, R2 is not a subset of R3.

**How many subspace does R2 have?**

How to Show that the Only Subspaces of R2 are the zero subspace, R2 itself, and the lines through the origin. I’m having trouble with a question from an introductory Linear Algebra book.