8. Denote by Rn the set of all n-tuples of real numbers. R is called the...

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8. Denote by Rn the set of all n-tuples of real numbers. R" is called the Euclidean vector space, with equality, addition and multiplication defined in the obvious way. Let V be the set of all vectors in R orthogonal to the vector (-1, 0, -1, 1); i.e. all vectors v E V so that v(-1, 0, -1, 1) = 0. (a) Prove that V is a subspace of R. (b) What is the dimension of V (provide an argument for this), and find a basis of V. (Hint: which 4-tuple does not belong to V for sure?). (c) Let A be a matrix consisting of the basis vectors of V (taken as rows of A). What is the Null-space of A and what is its row-space? Does the rank-nullity theorem hold for this matrix? 8. Denote by Rn the set of all n-tuples of real numbers. R" is called the Euclidean vector space, with equality, addition and multiplication defined in the obvious way. Let V be the set of all vectors in R orthogonal to the vector (-1, 0, -1, 1); i.e. all vectors v E V so that v(-1, 0, -1, 1) = 0. (a) Prove that V is a subspace of R. (b) What is the dimension of V (provide an argument for this), and find a basis of V. (Hint: which 4-tuple does not belong to V for sure?). (c) Let A be a matrix consisting of the basis vectors of V (taken as rows of A). What is the Null-space of A and what is its row-space? Does the rank-nullity theorem hold for this matrix?