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8)

[itex]U=\{x=(x_{1},x_{2},x_{3},x_{4})\in R^{4}|x_{1}+x_{2}+x_{4}=0\}[/itex]

is a subspace of [itex]R^{4}[/itex]

[itex]v=(2,0,0,1)\in R^{4}[/itex]

find [itex]u_{0}\in U[/itex] so [itex]||u_{0}-v||<||u-v||[/itex]

how i tried:

[itex]U=sp\{(-1,1,0,0),(-1,0,0,1),(0,0,1,0)\}[/itex]

i know that the only [itex]u_{0}[/itex] for which this innequality will work

is if it will be the orthogonal projection on U paralel to v

i am not sure about the theory of finding it

what to do next?

[itex]U=\{x=(x_{1},x_{2},x_{3},x_{4})\in R^{4}|x_{1}+x_{2}+x_{4}=0\}[/itex]

is a subspace of [itex]R^{4}[/itex]

[itex]v=(2,0,0,1)\in R^{4}[/itex]

find [itex]u_{0}\in U[/itex] so [itex]||u_{0}-v||<||u-v||[/itex]

how i tried:

[itex]U=sp\{(-1,1,0,0),(-1,0,0,1),(0,0,1,0)\}[/itex]

i know that the only [itex]u_{0}[/itex] for which this innequality will work

is if it will be the orthogonal projection on U paralel to v

i am not sure about the theory of finding it

what to do next?

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