Q1Multiple correct3 Marks4 Aug 2024Two vector spaces are isomorphic if there exists an isomorphism between them. Which vector spaces are isomorphic to R2\mathbb{R}^2R2?AV1={(x,y,z)∈R3∣x+y+z=0}V_1=\{(x,y,z)\in\mathbb{R}^3\mid x+y+z=0\}V1={(x,y,z)∈R3∣x+y+z=0}.BV2=span{(1,0,1),(1,1,0),(0,1,1)}V_2=\operatorname{span}\{(1,0,1),(1,1,0),(0,1,1)\}V2=span{(1,0,1),(1,1,0),(0,1,1)}.CV3={[abcd]∣a+b=0, c+d=0, a,b,c,d∈R}V_3=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\mid a+b=0,\ c+d=0,\ a,b,c,d\in\mathbb{R}\right\}V3={[acbd]∣a+b=0, c+d=0, a,b,c,d∈R}.DV4={x∈R3∣Ax=0, A=[2−11−1−71120]}V_4=\{x\in\mathbb{R}^3\mid Ax=0,\ A=\begin{bmatrix}2&-1&1\\-1&-7&1\\1&2&0\end{bmatrix}\}V4={x∈R3∣Ax=0, A=2−11−1−72110}.EV5V_5V5 is the column space of [6−321]\begin{bmatrix}6&-3\\2&1\end{bmatrix}[62−31].
