Q18Multiple correct4 Marks29 Oct 2023

Unless otherwise stated, assume that we consider the usual vector addition and scalar multiplication. Choose the correct option(s) from the following statements.

A

The set of vectors $\{(a,b)\in\mathbb{R}^2\}$ with scalar multiplication defined by $k(a,b)=(0,kb)$ forms a vector space.

B

The set of real numbers with addition defined by $x+y:=x-y$ forms a vector space.

C

Let $A\in\mathbb{R}^{3\times3}$ be an invertible matrix. The set of all solutions of the homogeneous system $AX=0$ is a vector space of dimension $0$.

D

Any vector subspace of $\mathbb{R}^2$ with dimension $1$ is of the form $ax+by=0$ where $a\ne0$ or $b\ne0$.

E

Any vector subspace of $\mathbb{R}^3$ with dimension $1$ is of the form $ax+by=c$ where $a\ne0$ or $b\ne0$.

F

$V=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}:a+b=c+d\right\}$ forms a vector space. (G) The set of all $n\times n$ matrices with rank strictly less than $n$ does not form a vector space.