How to find the perpendicular to the victor space R^k
To find the perpendicular to the vector space R^k, you first need to understand what it means for two vectors to be perpendicular. Two vectors are perpendicular if their dot product is zero.
In R^k, a vector can be represented as an ordered set of k real numbers, also known as coordinates. Let's assume we have a vector v = (v1, v2, ..., vk) in R^k.
To find the perpendicular vector, you can consider the orthogonal complement or orthogonal subspace. The orthogonal complement of a vector space V is the set of all vectors that are perpendicular to every vector in V. In this case, since we are dealing with R^k, the orthogonal complement is denoted as V⊥.
To find V⊥ for R^k, you can start by choosing a basis for R^k. A basis is a set of vectors that span the vector space. In this case, the standard basis for R^k is the set of k standard basis vectors: e1 = (1, 0, 0, ..., 0), e2 = (0, 1, 0, ..., 0), ..., ek = (0, 0, ..., 0, 1).
Next, for each basis vector, compute its dot product with the vector v. If the dot product is zero, then the vector is perpendicular to the given vector. Repeat this process for each basis vector to obtain all the vectors that are perpendicular to v.
By combining these perpendicular vectors, you can obtain the orthogonal complement V⊥.