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) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

14 Feb 2025

Answer :

Word Count : 526

Let's break down each statement one by one: --- i) If \(W_1\) and \(W_2\) are proper subspaces of a non-zero, finite-dimensional vector space \(V\) and \(\dim(W_1) > \dim(V)/2\), \(\dim(W_2) > \dim(V)/2\), then \(W_1 \cap W_2 \neq \{0\}\). - True. If \(W_1\) and \(W_2\) are proper subspaces of \(V\), then \(\dim(W_1) < \dim(V)\) and \(\dim(W_2) < \dim(V)\). Given that both \(\dim(W_1)\) and \(\dim(W_2)\) are greater than half the dimension of \(V\), their sum \(\dim(W_1) + \dim(W_2)\) is greater than \(\dim(V)\). This forces the intersection \(W_1 \cap W_2\) to contain at least one non-zero vector, meaning \(W_1 \cap W_2 \neq \{0\}\). --- ii) If \(V\) is a ____ _________ ________ _____ ______ __________.
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