Q6Multiple correct5 Marks13 Jul 2025

Each of the options below contains a set $V$ along with an addition operation $+:V\times V\to V$ and a scalar multiplication operation $\cdot:\mathbb{R}\times V\to V$. Choose the options for which there is an additive identity in $V$ with respect to $+$.

A

$V=\{x\in\mathbb{R}\mid x>0\}$, $x+y=xy$, and $c\cdot x=x^c$, for all $x,y\in V$ and $c\in\mathbb{R}$.

B

$V=\{(x,1)\mid x\in\mathbb{R}\}$, $(x,1)+(y,1)=(x+y,1)$, and $c\cdot(x,1)=(cx,1)$, for all $x,y,c\in\mathbb{R}$.

C

For an $m\times n$ matrix $A$ and a non-zero vector $b$ in its column space, $V=\{x\in\mathbb{R}^n\mid Ax=b\}$, and $+$, $\cdot$ denote the usual addition and scalar multiplication of vectors in $\mathbb{R}^n$, respectively.

D

$V$ is the set of skew-symmetric matrices of order $n$, i.e., $V=\{A\in M_{n\times n}(\mathbb{R})\mid A^T=-A\}$, and $+$, $\cdot$ denote the usual addition and scalar multiplication of $n\times n$ matrices, respectively.