Question

Show that if W is a subspace of Rnand vector v is in both W and W⊥ , then vector v μst be the zero vector.

a) Linear independence
b) Orthogonal complement
c) Linear transformation
d) Eigenvalues

Answer 1

Vector v must be the null vector (zero vector) if it belongs to both a subspace W and its orthogonal complement W¹, as it has to be orthogonal to itself, which only holds true for a vector with all components equal to zero.

Step-by-step explanation:

To show that if vector v is in both W and W¹, it must be the zero vector, we rely on the definition of the orthogonal complement. By definition, if vector v is in W¹, it means that vector v is orthogonal (perpendicular) to every vector in W, including itself when it is also part of W. Since the scalar product (dot product) of orthogonal vectors is zero, and vector v's scalar product with itself would only be zero if all its components are zero, vector v must be the null vector, with all components equal to zero.

The null vector is the unique vector in vector space which has no magnitude (length) and no specific direction, denoted by 0. Thus, if a vector v is in both a subspace and its orthogonal complement, it must be the case that the vector is the null vector since that is the only vector that is orthogonal to itself.