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Please can you guys help me to solve the following questions
Q1.if Z is an (n-1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null space of suitable linear functional f on X, which is uniquely determined to within a scalar multiple.
Use the exact same logic as my other post. Take a basisQuote:
Originally Posted by kinkong
for
and extend it to a basis
for
and define
by
. To prove it's unique up to a scaling factor note that any linear functional
must have the same form except
may be different and so evidently
note that the linear functional φ in this case is the dual vector to the basis element xn.
note also that since n = dim(X) = null(φ) + rank(φ) = null(φ) + 1, dim(null(φ)) = n-1.
it is a theorem that every hyper-space (or hyper-plane) arises in this fashion.