91Ó°ÊÓ

Vector addition is a fundamental operation in vector spaces, including subspaces. It involves adding two vectors together to produce a third vector. Imagine you have two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), both belonging to a certain set \( S \). For \( S \) to be considered a subspace, their sum \( \mathbf{a} + \mathbf{b} \) must also be in \( S \).

In our case, the set \( S \) consists of vectors of the form \( \begin{bmatrix} x \ x \ x \end{bmatrix} \). This means that for any two vectors in \( S \), say \( \mathbf{u} = \begin{bmatrix} x \ x \ x \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} y \ y \ y \end{bmatrix} \), their sum is \( \mathbf{u} + \mathbf{v} = \begin{bmatrix} x+y \ x+y \ x+y \end{bmatrix} \). This resulting vector is still of the form \( \begin{bmatrix} z \ z \ z \end{bmatrix} \), showing that it remains in \( S \).

• This confirms the subspace property of being closed under addition.
• Vector addition ensures that the structure of a subspace is maintained when vectors within it are combined.

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). It is another vital operation for any subspace within a vector space. For a set to qualify as a subspace, it must be closed under scalar multiplication.

When focusing on our example with set \( S \), each vector is of the form \( \begin{bmatrix} x \ x \ x \end{bmatrix} \). When you multiply this vector by a scalar \( c \), the result is \( c \begin{bmatrix} x \ x \ x \end{bmatrix} = \begin{bmatrix} cx \ cx \ cx \end{bmatrix} \). Since this new form \( \begin{bmatrix} z \ z \ z \end{bmatrix} \) remains within \( S \), closure under scalar multiplication is verified.

• This property shows that every scalar multiple of a vector in the subspace is still part of the subspace.
• Scalar multiplication helps to scale vectors up or down while retaining their alignment within the subspace.

The zero vector is a unique vector where all components are zero, such as \( \mathbf{0} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix} \) in \( \mathbb{R}^3 \). A subspace must contain the zero vector to adhere to its mathematical definition.

In our example, ensuring the zero vector is part of set \( S \) involves substituting zero for all components in the vectors of the form \( \begin{bmatrix} x \ x \ x \end{bmatrix} \). Doing so yields \( \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix} \), which is precisely the zero vector required in \( S \).

• Inclusion of the zero vector is crucial because it represents the additive identity in vector spaces.
• It establishes the baseline for other vectors, ensuring that they can return to a starting, neutral point through operations like addition.