Which of the following statements are true and which are false? Justif

Question

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) The function $f:R\rightarrow R$ defined by f(x) = cosx is 1-1.

ii) The operation * defined by x*y = log(xy) is a binary operation on S, where S is the set {x ∈ R x > 0}.

(iii)The set {(x₁.x₂,...,xn) X₁,X₂,...,xn ∈ R,X₁ = 2x2+3} is a subspace of Rⁿ

iv) There is no 7 x 5 matrix of rank 6.

v) If V and V' are vector spaces and T: V→ V' is a linear transformation, then whenever u₁,u₂,..., u_k are linearly independent, T_u₁, T_u2, ... Tu_kare also linearly independent.

vi) If V is a vector space and T: V → V is a linear operator with det(T) = 0, then T is not diagonalisable.

vii) The degree of the minimal polynomial of a 3 x 3 matrix is at most 2.

viii) For any 2 x 2 matrix A, Adj (A') = (Adj(A))'.

ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.

x) There is no co-ordinate transformation that transforms the quadratic form x²+y² +z² to the quadratic form xz+yz.

17 Feb 2025

Answer :

Word Count : 756

Let's address each of these statements: i) The function $ f: \mathbb{R} \to \mathbb{R} $ defined by $ f(x) = \cos(x) $ is 1-1. - False. - A function is one-to-one (injective) if every element of the range is mapped from a distinct element of the domain. The cosine function is periodic with a period of $ 2\pi $, meaning that for $ x $ and $ x + 2\pi $, $ f(x) = f(x + 2\pi) $. For example, $ f(0) = f(2\pi) = 1 $, so the function is not injective. ii) The operation * defined by $ x * y = \log(xy) $ is a binary operation on $ S $, where $ S = \{ x \in \mathbb{R} : x > 0 \} $. - True. - A binary operation requires closure, associativity, and the operation must be well-defined. For $ x, y \in S $, $ x * y = \log(xy) $ is well-defined because the product $ xy $ is positive, and the logarithm of a positive number is defined. The operation is closed because $ \log(xy) $ is real for all \( ____ ____ ____ ________ _______ _______ ___ __________.
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