91Ó°ÊÓ

To understand subspaces, think of them as smaller spaces within a larger vector space. For instance, if we have a vector space \( \mathbb{R}^n \), a subspace of \( \mathbb{R}^n \) is a subset of \( \mathbb{R}^n \) that is also a vector space itself. This means that a subspace must satisfy three important conditions:

• It must contain the zero vector.
• It must be closed under vector addition (if you take any two vectors in the subspace, their sum is also in the subspace).
• It must be closed under scalar multiplication (if you multiply any vector in the subspace by a scalar, the result is still in the subspace).

In the context of the problem, \( V \) and \( W \) are subspaces of \( \mathbb{R}^n \) and \( \mathbb{R}^m \), respectively. This means they fulfill all the conditions listed above within their respective larger spaces.

A basis of a subspace is a set of vectors that are both linearly independent and span the subspace. Let's break this down into simpler terms.

• Linearly Independent: This means that no vector in the set can be written as a linear combination of the others. For example, in the set \(\{\vec{v}_1, \vec{v}_2\}\), if \( \vec{v}_2 \) could be expressed as \(c_1\vec{v}_1\), then \(\{ \vec{v}_1, \vec{v}_2 \}\r\) would not be linearly independent.
• Span: This means that any vector in the subspace can be written as a linear combination of the basis vectors. If \(\{ \vec{v}_1, \vec{v}_2 \} \) spans the subspace, then any vector in that subspace can be written as \(a\vec{v}_1 + b\vec{v}_2\).
  

In our problem, \( \{\vec{v}_1, \ldots, \vec{v}_r\}\r\) is a basis for \(V\r\). So, any vector in \(V\r\) can be written as a linear combination of these basis vectors.

`Span` refers to all possible linear combinations of a set of vectors. If you take a set of vectors and combine them in every possible way with various scalars, the set of all these combinations is the `span` of that set.
For example, the span of the set \{ \vec{v}_1, \vec{v}_2 \} is the set of all vectors that can be written as \(c_1\vec{v}_1 + c_2\vec{v}_2\) where \(c_1\) and \(c_2\) are scalars.

• If the span of the set is the entire space, the vectors are said to span the space.
• If the span is just a subspace, then the vectors span that subspace.

In our problem, we've shown that \( \{ T(\vec{v}_1), \cdots, T(\vec{v}_r) \} \r\) spans \(W\r\). This means every vector in \(W\r\) can be written as a linear combination of \( \{ T(\vec{v}_1), \cdots, T(\vec{v}_r) \}\r\). Hence, the span of these transformed vectors covers all of \(W\r\).