Linear Algebra Question Let V be the set of 2 by 2 invertibl

Linear Algebra Question

Let V be the set of 2 by 2 invertible matrices. Given two elements A and B of V, and c epsilon R. we define the operations A B = A.B and c times A = c middot A. where A.B represents usual matrix multiplication and c middot A represents usual scalar multiplication of matrices. We will refer to as addition and times as scalar multiplication on V. Is the set V closed under ? Does there exist a 0-vector in V relative to ? Do there exist additive inverses relative to ? For each question, briefly explain your answer, do not simply write yes/no. Show that (V, x) is not a vector space by providing explicit examples where each of the following 3 axioms fail: Closure under scalar multiplication Commutativity of addition Distribution of scalar multiplication ((r + s) times A = (r times A) (s times A)) If S = {v_1, v_2, .., v_n} is a subset of a vector space H\', and u epsilon span S, prove that the set {u, v_1, .., v_n} is linearly dependent.

Solution

a) Since the product of two invertible matrices is invertible. So V is closed under the given operation.

The Zero vector of V is I the identity matrix of order 2. As IA=AI=A

The additive inverse with respect to the given operation is A^{-1}. Which exists in V .

b) if c=0 then c•A=0 which is not incvertible.

Also product of two matrices is not commutative. So the given operation is not commutative.