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

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 → R defined by f(x) = cos x 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 $\left \{ \left ( x_1.x_2,....,x_n \right )|x_1,x_2,...,x_n \right \}\epsilon R,x_1=2x\}$ is a subspace of $R^{n}.$

.
iv) There is no 7×5 matrix of rank 6.
v) If $V{}'$ and V 0 are vector spaces and T : V → V 0 is a linear transformation, then
whenever u1,,...,uk are linearly independent, Tu1, Tu2, ..., Tuk are 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 3×3 matrix is at most 2.
viii) For any 2×2 matrix $A, Adj(A^t)=(Adj(A))^t.$
.
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 $x^2+y^2+z^2$ to the quadratic form xz+yz.

11 Mar 2024

Answer :

Word Count : 706

Here are the answers with justifications: ### (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 (1-1) if $ f(x_1) = f(x_2) $ implies $ x_1 = x_2 $. However, $ \cos x $ is periodic with period $ 2\pi $, meaning $ \cos x = \cos(x + 2\pi k) $ for any integer $ k $. This contradicts injectivity. --- ### (ii) The operation $ \ast $ defined by $ x \ast y = \log(xy) $ is a binary operation on $ S $, where $ S = \{ x \in \mathbb{R} | x > 0 \} $. True A binary operation on $ S $ means that for all $ x, y \in S $, $ x \ast y $ must also be __________ ______ _____ ______ ____ _______ __________.
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