Assuming I have a vector $v_1 \in \mathbb{R}^n $ of dimension $n$. (edit what I mean is e.g. $v_1 \in \mathbb{R}^n $)
I want to describe the set of "all the vectors that are orthogonal to $v_1$".
- Would it be correct to call this set the "null space of $v_1$"?
- Would "orthogonal complement of $V=\{v_1\}$ in
the whole vector space of dimension n$\mathbb{R}^n$" be a better / more correct definition?
Suppose I have the set of "all the vectors that are orthogonal to $v_1$" at hand and I also found a basis $B$ for this set: $$B = \{b_1, b_2,…, b_{n-1}\}$$
And suppose further that another vector $n$-dimensional vector $v_2 \in \mathbb{R}^n$, $v_2 \neq v_1 $ is expressed in this basis $B$: $$[v_2]_B = \sum_{k=0}^{n-1}b_k<b_k,v_2>$$
- What would be an intuitive explanation for the relationship between $v_1$ and $[v_2]_B$? Did this process of obtaining $[v_2]_B$ somehow "remove traces" of $v_1$ in $v_2$? Is there any intuition for this?
Thanks in advance!
EDIT
Sorry some further question:
- If $v_1 = v_2$, then $[v_2]_B = 0$. So can $|[v_2]_B|$ be seen as some sort of similarity measurement?
Best Answer
An $m \times n$ matrix $M$ represents a linear map from $R^n$ to $R^m$ in the standard bases of these spaces. As such, if a vector $v=(v_1,...,v_n)$ is turned into the $1 \times n$ matrix $M=[v_1,v_2,...,v_n]$ then $M$ represents a linear map from $R^n$ to $R^1$, and the null space of $M$ is the same as the set of vectors $w$ for which $v \cdot w=0$.
Another term used for this is the orthogonal complement of the vector $v$ in $R^n$.