91Ó°ÊÓ

$\mathrm{T}\left(\mathrm{q}({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)\right)=\mathrm{q}({\mathrm{x}}_1,1,1)$

$$\begin{gathered} \mathrm{Consider},{\mathrm{q}}_1\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)={\mathrm{a}}_1{\mathrm{x}}_1^2+{\mathrm{b}}_1{\mathrm{x}}_2^2+{\mathrm{c}}_1{\mathrm{x}}_3^2+{\mathrm{d}}_1{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{e}}_1{\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{f}}_1{\mathrm{x}}_3{\mathrm{x}}_1, \\ {\mathrm{q}}_1\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)={\mathrm{a}}_1{\mathrm{x}}_1^2+{\mathrm{b}}_1{\mathrm{x}}_2^2+{\mathrm{c}}_1{\mathrm{x}}_3^2+{\mathrm{d}}_1{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{e}}_1{\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{f}}_1{\mathrm{x}}_3{\mathrm{x}}_1,∈{\mathrm{Q}}_3\mathrm{and}\mathrm{Î\pm }∈\mathrm{R},\mathrm{then} \\ {\mathrm{a}}_2{\mathrm{x}}_1^2+{\mathrm{b}}_2{\mathrm{x}}_2^2+{\mathrm{c}}_2{\mathrm{x}}_3^2+{\mathrm{d}}_2{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{e}}_1{\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{f}}_1{\mathrm{x}}_3{\mathrm{x}}_1) \\ =\mathrm{T}\left(\left({\mathrm{Î\pm »å}}_1+{\mathrm{d}}_2\right){\mathrm{x}}_1^2+\left({\mathrm{Î\pm ²ú}}_1+{\mathrm{b}}_2\right){\mathrm{x}}_2^2+({\mathrm{Î\pm ³¦}}_1+{\mathrm{c}}_2\right){\mathrm{x}}_3^2+ \\ \left({\mathrm{Î\pm »å}}_1+{\mathrm{d}}_2\right){\mathrm{x}}_1{\mathrm{x}}_2+\left({\mathrm{Î\pm \pm ð}}_1+{\mathrm{e}}_2\right){\mathrm{x}}_2{\mathrm{x}}_3+({\mathrm{Î\pm ´Ú}}_1+{\mathrm{f}}_2){\mathrm{x}}_3{\mathrm{x}}_1 \\ =\left({\mathrm{Î\pm »å}}_1+{\mathrm{a}}_2\right){\mathrm{x}}_1^2+({\mathrm{Î\pm ²ú}}_1+{\mathrm{b}}_2+({\mathrm{Î\pm ³¦}}_1+{\mathrm{c}}_2)+ \\ \left({\mathrm{Î\pm »å}}_1+{\mathrm{d}}_2\right){\mathrm{x}}_1+\left({\mathrm{Î\pm \pm ð}}_1+{\mathrm{e}}_2\right)+({\mathrm{Î\pm ´Ú}}_1+{\mathrm{f}}_2) \\ =\mathrm{Î\pm }\left({\mathrm{a}}_1{\mathrm{x}}_1^2+{\mathrm{b}}_1+{\mathrm{c}}_1+{\mathrm{d}}_1{\mathrm{x}}_1+{\mathrm{e}}_1{\mathrm{f}}_1{\mathrm{x}}_1\right)+\left({\mathrm{a}}_2{\mathrm{x}}_1^2+{\mathrm{b}}_2+{\mathrm{c}}_2+{\mathrm{d}}_2{\mathrm{x}}_1+{\mathrm{e}}_2{\mathrm{f}}_2{\mathrm{x}}_1\right) \\ ={\mathrm{Î\pm \pm ç}}_1\left({\mathrm{x}}_1,1,1\right)+{\mathrm{q}}_2\left({\mathrm{x}}_1,1,1\right) \\ =\mathrm{Î\pm °Õ}\left({\mathrm{q}}_1\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\right)+\mathrm{T}\left({\mathrm{q}}_2\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\right) \end{gathered}$$,

Since satisfy

$$\begin{gathered} \mathrm{T}\left({\mathrm{Î\pm \pm ç}}_1+{\mathrm{q}}_2\right)=\mathrm{Î\pm °Õ}\left({\mathrm{q}}_1\right)+\mathrm{T}\left({\mathrm{q}}_2\right)\mathrm{for}\mathrm{Î\pm }∈\mathrm{R}\mathrm{and}{\mathrm{q}}_1,{\mathrm{q}}_2{\mathrm{iQ}}_{3,} \\ \mathrm{so}\mathrm{T}:{\mathrm{Q}}_3→{\mathrm{P}}_2,\mathrm{T}\left(\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\right)=\mathrm{q}\left({\mathrm{x}}_{1}1,1\right) \\ \mathrm{Is}\mathrm{a}\mathrm{linear}\mathrm{transformation}. \end{gathered}$$

$$\begin{gathered} \ker \left(\mathrm{T}\right)=\left\{\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)∈{\mathrm{Q}}_3:\mathrm{T}\left(\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\right)=0∈{\mathrm{P}}_2\right\} \\ =\left\{\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)∈{\mathrm{Q}}_2:\mathrm{q}\left({\mathrm{x}}_1,1,1\right)=0∈{\mathrm{P}}_2\right\} \\ =\left\{{\mathrm{ax}}_1^2+{\mathrm{bx}}_2^2+{\mathrm{dx}}_1{\mathrm{x}}_2+{\mathrm{ex}}_2{\mathrm{x}}_3+{\mathrm{fx}}_3{\mathrm{x}}_1∈{\mathrm{Q}}_3:\mathrm{q}\left({\mathrm{x}}_1,1,1\right)=0\right\} \\ =\left\{\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)∈{\mathrm{Q}}_3:{\mathrm{ax}}_1^2\left(\mathrm{d}+\mathrm{f}\right){\mathrm{x}}_1+\left(\mathrm{b}+\mathrm{c}+\mathrm{e}\right)=0∈{\mathrm{P}}_2\right\} \\ =\left\{{\mathrm{ax}}_1^2+{\mathrm{bx}}_2^2+{\mathrm{cx}}_3^2+{\mathrm{dx}}_1{\mathrm{x}}_2+{\mathrm{ex}}_2{\mathrm{x}}_3+{\mathrm{fx}}_3{\mathrm{x}}_1∈{\mathrm{Q}}_3:\mathrm{a}=0,\mathrm{f}=-\mathrm{b}-\mathrm{c}\right\} \\ =\left\{\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)={\mathrm{bx}}_2^2+{\mathrm{cx}}_3^2+{\mathrm{dx}}_1{\mathrm{x}}_2-\left(\mathrm{b}+\mathrm{c}\right){\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{dx}}_3{\mathrm{x}}_1:\mathrm{b},\mathrm{c},\mathrm{d}∈\mathrm{R}\right\} \\ \mathrm{lm}\left(\mathrm{T}\right)=\left\{\mathrm{T}\left(\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\right)∈{\mathrm{P}}_2:\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)∈{\mathrm{Q}}_3\right\} \\ =\left\{\mathrm{q}\left({\mathrm{x}}_1,1,1\right)∈{\mathrm{P}}_2:\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)={\mathrm{ax}}_1^2+{\mathrm{bx}}_3^2+{\mathrm{dx}}_1{\mathrm{x}}_2+{\mathrm{ex}}_1{\mathrm{x}}_2+{\mathrm{fx}}_3{\mathrm{x}}_1∈{\mathrm{Q}}_3\right\} \\ =\left\{\mathrm{q}\left({\mathrm{x}}_1,1,1\right)={\mathrm{a}}_1{\mathrm{x}}_1^2+\left(\mathrm{d}+\mathrm{f}\right){\mathrm{x}}_1+\left(\mathrm{b}+\mathrm{c}+\mathrm{e}\right)∈{\mathrm{P}}_2\right\} \\ \left\{\mathrm{p}\left(\mathrm{x}\right)={\mathrm{Î\pm ³\ae }}^2+\mathrm{β³\ae }+\mathrm{y}∈{\mathrm{P}}_2:\mathrm{Î\pm }=\mathrm{a},\mathrm{β}=\mathrm{d}+\mathrm{f},\mathrm{y}=\mathrm{b}+\mathrm{c}+\mathrm{e}∈\mathrm{R}\right\} \\ \mathrm{rank}\left(\mathrm{T}\right)=\dim \left(\mathrm{lm}\left(\mathrm{T}\right)\right)=3 \\ \mathrm{nullity}\left(\mathrm{T}\right)=\dim \left(\ker \left(\mathrm{T}\right)\right)=4 \end{gathered}$$

Yes,T is a linear transformation

$$\begin{gathered} \ker \left(\mathrm{T}\right)=\left\{\mathrm{q}\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)={\mathrm{bx}}_2^2+{\mathrm{cx}}_3^2+{\mathrm{dx}}_1{\mathrm{x}}_2-\left(\mathrm{b}+\mathrm{c}\right){\mathrm{x}}_2{\mathrm{x}}_3-{\mathrm{dx}}_3{\mathrm{x}}_1∈{\mathrm{Q}}_3\right\} \\ \mathrm{lm}\left(\mathrm{T}\right)=\left\{\mathrm{p}\left(\mathrm{x}\right)={\mathrm{Î\pm ³\ae }}^2+\mathrm{β³\ae }+\mathrm{λ}∈{\mathrm{P}}_2\right\} \\ \mathrm{rank}\left(\mathrm{T}\right)=3\mathrm{and}\mathrm{unllity}\left(\mathrm{T}\right)=3 \end{gathered}$$