To find the dimension of the subspace [tex]\( W \)[/tex] defined by [tex]\( W = \left\{ x \in \mathbb{R}^4 : v^T x = 0 \right\} \)[/tex], where [tex]\( v = \begin{bmatrix} 1 \\ 2 \\ -3 \\ -1 \end{array} \)[/tex], we need to understand the concept of orthogonal subspaces.
Given a vector [tex]\( v \)[/tex] in [tex]\( \mathbb{R}^4 \)[/tex], [tex]\( W \)[/tex] consists of all vectors [tex]\( x \)[/tex] in [tex]\( \mathbb{R}^4 \)[/tex] that are orthogonal to [tex]\( v \)[/tex].
To find the dimension of [tex]\( W \)[/tex]:
1. Identify the space:
- The vector [tex]\( v \)[/tex] belongs to [tex]\( \mathbb{R}^4 \)[/tex].
2. Orthogonal subspace:
- The orthogonal subspace [tex]\( W \)[/tex] is defined by all vectors in [tex]\( \mathbb{R}^4 \)[/tex] that are orthogonal to [tex]\( v \)[/tex].
3. Dimensionality concept:
- If [tex]\( v \)[/tex] is a non-zero vector in [tex]\( \mathbb{R}^n \)[/tex], the subspace of vectors orthogonal to [tex]\( v \)[/tex] (i.e., [tex]\( W \)[/tex]) is spanned by [tex]\( n-1 \)[/tex] dimensions.
- This is because the vector [tex]\( v \)[/tex] itself reduces the dimensionality of the space by 1 since the orthogonal vectors lie in a plane orthogonal to [tex]\( v \)[/tex].
4. Apply the concept:
- Here, the vector [tex]\( v \)[/tex] is in [tex]\( \mathbb{R}^4 \)[/tex].
- Therefore, the dimension of the subspace [tex]\( W \)[/tex], which is orthogonal to [tex]\( v \)[/tex], will be:
[tex]\[
\operatorname{dim}(W) = 4 - 1 = 3
\][/tex]
Thus, the dimension of [tex]\( W \)[/tex] is [tex]\( 3 \)[/tex].
