# Proving W is a subspace.

• dylanhouse
In summary, a subspace is a subset of a vector space that has all the properties of the original vector space, such as closure under addition and scalar multiplication, and containing the zero vector. To prove that W is a subspace, we need to show that it satisfies the three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector. Closure under addition means that the resulting vector after adding two vectors in W must also be in W, ensuring that the subspace remains within the original vector space. Closure under scalar multiplication means that the resulting vector after multiplying a vector in W by a scalar must also be in W, ensuring that the subspace is closed under all possible scalar operations. To show that W contains the

## Homework Statement

Given W={A belonging to M2(ℂ) | A is symmetric} is a subspace of M2(ℂ) over ℂ, when showing it is closed under scalar multiplication, do I need to use a complex scalar as it is over the complex numbers, or will a real number be okay?

## The Attempt at a Solution

You have to prove that it's closed under scalar multiplication using complex numbers, as the vector space $M_2(\mathbb{C})$ is a vector space over the complex numbers.

## What is a subspace?

A subspace is a subset of a vector space that contains all the properties of the original vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.

## How do you prove that W is a subspace?

To prove that W is a subspace, we need to show that it satisfies the three conditions: closure under addition, closure under scalar multiplication, and contains the zero vector. This can be done by using the definition of a subspace and showing that all the properties hold for the elements in W.

## What is closure under addition?

Closure under addition means that when we add two vectors in W, the resulting vector must also be in W. This ensures that the subspace remains within the original vector space and does not contain any elements that do not belong to it.

## What is closure under scalar multiplication?

Closure under scalar multiplication means that when we multiply a vector in W by a scalar, the resulting vector must also be in W. This ensures that the subspace is closed under all possible scalar operations and does not contain any elements that do not belong to it.

## How do you show that W contains the zero vector?

To show that W contains the zero vector, we need to demonstrate that the zero vector is an element of W. This can be done by showing that the zero vector satisfies all the properties of the subspace, such as closure under addition and scalar multiplication.