A subspace of a vector space is a non-empty subset that is closed under vector addition and scalar multiplication. This means that if two vectors are in the subspace, then their sum and scalar multiples are also in the subspace.

The set of all vectors in $\mathbb{R}^3$ with zero as their first coordinate is a subspace of $\mathbb{R}^3$ because it is non-empty, closed under vector addition, and closed under scalar multiplication.

Subspaces are not always finite-dimensional. For example, the subspace of $\mathbb{R}^3$ consisting of all vectors with zero as their first coordinate is infinite-dimensional.

This is one of the properties of subspaces. If $\mathbf{u}$ and $\mathbf{v}$ are in $W$, then $\mathbf{u} + \mathbf{v}$ is also in $W$ because $W$ is closed under vector addition.

This is another property of subspaces. If $\mathbf{u}$ is in $W$, then $c\mathbf{u}$ is also in $W$ because $W$ is closed under scalar multiplication.

The intersection of two subspaces is a subspace because it is non-empty, closed under vector addition, and closed under scalar multiplication.

The union of two subspaces is not always a subspace. For example, the union of the subspace of $\mathbb{R}^3$ consisting of all vectors with zero as their first coordinate and the subspace of $\mathbb{R}^3$ consisting of all vectors with zero as their second coordinate is not a subspace of $\mathbb{R}^3$.

The set of all solutions to a system of linear equations is a subspace because it is non-empty, closed under vector addition, and closed under scalar multiplication.

The set of all polynomials of degree less than or equal to $n$ is a subspace because it is non-empty, closed under vector addition, and closed under scalar multiplication.

The set of all matrices of order $m \times n$ is a subspace because it is non-empty, closed under matrix addition, and closed under scalar multiplication.

Every subspace of a vector space is a vector space because it inherits the vector space operations and properties from the larger vector space.

The dimension of a subspace of a vector space is always less than or equal to the dimension of the vector space because the subspace is a subset of the vector space.

The zero subspace of a vector space is the subspace consisting of the zero vector only because it is non-empty, closed under vector addition, and closed under scalar multiplication.

The trivial subspace of a vector space is the subspace consisting of the zero vector only, not all vectors in the vector space.

The span of a set of vectors in a vector space is the smallest subspace of the vector space that contains the set of vectors because it is the intersection of all subspaces that contain the set of vectors.