91Ó°ÊÓ

In linear algebra, a vector space, also known as a linear space, is a fundamental concept that provides a framework for understanding vectors. A vector space consists of a set of vectors along with two operations: vector addition and scalar multiplication.

A vector space must satisfy several properties for these operations:

• **Commutative Law for Addition**: For any two vectors \( \mathbf{u} \) and \( \mathbf{v} \) in the space, \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
• **Associative Law for Addition**: For any vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
• **Existence of Zero Vector**: There is a vector, often written as \( \mathbf{0} \), such that \( \mathbf{u} + \mathbf{0} = \mathbf{u} \) for any vector \( \mathbf{u} \).
• **Existence of Additive Inverses**: For every vector \( \mathbf{u} \), there is a vector \( -\mathbf{u} \) such that \( \mathbf{u} + (-\mathbf{u}) = \mathbf{0} \).
• **Distributive Laws for Scalar Multiplication**: \( a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v} \) and \( (a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u} \) for scalars \(a\) and \(b\).
• **Compatibility of Scalar Multiplication**: \( a(b\mathbf{u}) = (ab)\mathbf{u} \).
• **Identity of Scalar Multiplication**: \( 1\mathbf{u} = \mathbf{u} \) for any vector \( \mathbf{u} \).

Understanding these fundamental properties will help you recognize vector spaces across different dimensions and contexts, including \( \mathbb{R}^3 \), the space of all 3-dimensional vectors.

Subspaces are a special type of vector spaces that exist within larger vector spaces. Simply put, a subspace is itself a vector space, closed under the same operations. To identify a subspace within a vector space like \( \mathbb{R}^3 \), it needs to satisfy a few key conditions:

• **Contains the Zero Vector**: The zero vector must be part of the subspace. This ensures non-emptiness.
• **Closure under Addition**: If you take any two vectors from the subspace and add them, the result must also be a vector within the same subspace.
• **Closure under Scalar Multiplication**: Multiplying any vector in the subspace by a scalar produces another vector that remains in the subspace.

Examining the given set \( W = \{ [5b+2c, b, c]^T \mid b, c \text{ are arbitrary} \} \) from the exercise, we verify these criteria. The presence of the zero vector is confirmed by choosing \( b = 0 \) and \( c = 0 \). Closure under addition and scalar multiplication follows naturally from the linear form of the vector expression, ensuring \( W \) is indeed a subspace of \( \mathbb{R}^3 \).

Understanding linear combinations is the heart of many vector space operations, including spanning sets, bases, and more. A linear combination involves creating a new vector using a set of vectors and multiplying each by a corresponding scalar coefficient. These products are then summed together.

Given the vectors:

• \( \mathbf{u} = \begin{bmatrix} 5 \ 1 \ 0 \end{bmatrix} \)
• \( \mathbf{v} = \begin{bmatrix} 2 \ 0 \ 1 \end{bmatrix} \)

Any vector in \( W \) can be expressed as a linear combination of \( \mathbf{u} \) and \( \mathbf{v} \) by defining the scalars as \( b \) and \( c \): \[ b\begin{bmatrix} 5 \ 1 \ 0 \end{bmatrix} + c\begin{bmatrix} 2 \ 0 \ 1 \end{bmatrix} = \begin{bmatrix} 5b + 2c \ b \ c \end{bmatrix} \]
This representation shows how every vector in \( W \) can be built using these selected vectors and appropriate scalar multipliers.

The concepts of basis and span are closely intertwined, forming the cornerstone of understanding vector spaces and subspaces. The **span** of a set of vectors is the collection of all possible linear combinations of those vectors. It effectively "brings together" any number of those basis vectors to form any vector that belongs in the space or subspace.

For the set \( W = \operatorname{Span}\{ \mathbf{u}, \mathbf{v} \} \), where \( \mathbf{u} = \begin{bmatrix} 5 \ 1 \ 0 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 2 \ 0 \ 1 \end{bmatrix} \):

• **Span**: Consists of all vectors that you can generate using linear combinations of \( \mathbf{u} \) and \( \mathbf{v} \).
• **Basis**: A set of vectors that not only spans a space but is also linearly independent. This means no vector in the set can be written as a linear combination of the others.

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) form a basis for \( W \). They span the subspace \( W \), and because neither vector can be expressed as a linear combination of the other, they are linearly independent. Consequently, they provide a foundation from which any vector in \( W \) can be constructed through linear combinations, showcasing the power of basis and span in vector space theory.