banganX

2020-11-07

Prove that

StrycharzT

Skilled2020-11-08Added 102 answers

Let $({x}_{1},{y}_{1}),({x}_{2},{x}_{2})\in {\mathbb{R}}^{2}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\le t{c}_{1},{c}_{2}\in \mathbb{R}$ . To prove that T is a linear transformation, we must prove that

$T({c}_{1}({x}_{1},{y}_{1})+{c}_{2}({x}_{2},{y}_{2}))={c}_{1}T({x}_{1},{y}_{1})+{c}_{2}T({x}_{2},{y}_{2})$

This is done by a direct computation:

$T({c}_{1}({x}_{1},{y}_{1})+{c}_{2}({x}_{2},{y}_{2}))$

$=T({c}_{1}{x}_{1},{c}_{2}{x}_{2})+({c}_{2}{x}_{2},{c}_{2}{x}_{2})$

$=T({c}_{1}{x}_{1}{c}_{2}{x}_{2},{c}_{1}{y}_{1}+{c}_{2}{y}_{2})$

$=(3({c}_{1}{x}_{1}+{c}_{2}{x}_{2})+({c}_{1}{y}_{1}+{c}_{2}{y}_{2}),2({c}_{1}{y}_{1}+{c}_{2}{y}_{2}),({c}_{1}{x}_{1}+{c}_{2}{x}_{2})-({c}_{1}{y}_{1}+{c}_{2}{y}_{2}))$

$=(3{c}_{1}{x}_{1}+3{c}_{2}{x}_{2}+{c}_{1}{y}_{1}+{c}_{2}{y}_{2},2{c}_{1}{y}_{1}+{c}_{2}{y}_{2},{c}_{1}{x}_{1}+{c}_{2}{x}_{2}-{c}_{1}{y}_{1}+{c}_{2}{y}_{2})$

$=(3{c}_{1}{x}_{1}+{c}_{1}{y}_{1},2{c}_{1}{y}_{}$

This is done by a direct computation:

12

22+64

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