Question 6 - Week 3 Practice | Prasnya
Each of the options below contains a set V V along with an addition operation + : V × V → V +:V\times V\to V and a scalar multiplication operation ⋅ : R × V → V \cdot:\mathbb{R}\times V\to V . Choose the options for which there is an additive identity in V V with respect to + + . Multiple correct
Hard Difficulty
13 July 2025
A V = { x ∈ R ∣ x > 0 } V=\{x\in\mathbb{R}\mid x>0\} , x + y = x y x+y=xy , and c ⋅ x = x c c\cdot x=x^c , for all x , y ∈ V x,y\in V and c ∈ R c\in\mathbb{R} . B V = { ( x , 1 ) ∣ x ∈ R } V=\{(x,1)\mid x\in\mathbb{R}\} , ( x , 1 ) + ( y , 1 ) = ( x + y , 1 ) (x,1)+(y,1)=(x+y,1) , and c ⋅ ( x , 1 ) = ( c x , 1 ) c\cdot(x,1)=(cx,1) , for all x , y , c ∈ R x,y,c\in\mathbb{R} . C For an m × n m\times n matrix A A and a non-zero vector b b in its column space, V = { x ∈ R n ∣ A x = b } V=\{x\in\mathbb{R}^n\mid Ax=b\} , and + + , ⋅ \cdot denote the usual addition and scalar multiplication of vectors in R n \mathbb{R}^n , respectively. D V V is the set of skew-symmetric matrices of order n n , i.e., V = { A ∈ M n × n ( R ) ∣ A T = − A } V=\{A\in M_{n\times n}(\mathbb{R})\mid A^T=-A\} , and + + , ⋅ \cdot denote the usual addition and scalar multiplication of n × n n\times n matrices, respectively. 