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TIFR Mathematics 2024 | Part B | Question: 15
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A map $f: V \rightarrow W$ between finite dimensional vector spaces over $\mathbb{Q}$ is a linear transformation if and only if $f(x)=f(x-a)+f(x-b)-f(x-a-b)$, for all $x, a, b \in V$.
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TIFR Mathematics 2024 | Part B | Question: 1
If $\text{G}$ is a group of order $361$, then $\text{G}$ has a normal subgroup $\text{H}$ such that $H \cong G / H$.
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TIFR Mathematics 2024 | Part B | Question: 2
There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.
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TIFR Mathematics 2024 | Part B | Question: 3
The function $d: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ given by $d(x, y)=\left|e^{x}-e^{y}\right|$ defines a metric on $\mathbb{R}$, and $(\mathbb{R}, d)$ is a complete metric space.
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TIFR Mathematics 2024 | Part B | Question: 4
Let $n$ be a positive integer, and $A$ an $n \times n$ matrix over $\mathbb{R}$ such that $A^{3}=\mathrm{Id}$. Then $A$ is diagonalizable in $\mathrm{M}_{n}(\mathbb{R})$, i.e., there exists $P \in \mathrm{M}_{n}(\mathbb{R})$ such that $P$ is invertible and $P A P^{-1}$ is a diagonal matrix.
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