Show that if ~w is orthogonal to ~u and ~v, then ~w is orthogonal to every vector ~x in Span{~u, ~v}.

Question

Show that if ~w is orthogonal to ~u and ~v, then ~w is orthogonal to every vector ~x in Span{~u, ~v}.

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Ivy 1 week 2021-10-12T08:04:55+00:00 1 Answer 0

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    2021-10-12T08:05:58+00:00

    Explanation:

     \overline{w} is ortogonal to a vector  \overline{c} if, and only if, the scalar product  < \overline{w},\overline{c} > = 0. Hence, it should be < \overline{w},\overline{u} > = < \overline{w},\overline{v} > = 0 .

    The scalar product is linear, so it takes constants and sums out. If  \overline{x} is a vector spanned by  \overline{u} and  \overline{w} ,  lets say  \overline{x} = a*\overline{u} + b*\overline{v} , for certain complex (or real) values a and b, then we have

     < \overline{w},\overline{x} > = < \overline{w}, a*\overline{u} + b*\overline{v}> = a * < \overline{w},\overline{u} > + b * < \overline{w},\overline{v} > = a*0+b*0 = 0

    Because both < \overline{w},\overline{u} > and < \overline{w},\overline{v} > are equal to 0. That proves that  \overline{x} , an arbitrary element in  Span\{\overline{u}, \overline{v}\} , is perpendicular to  \overline{w} .

    I hope that helped you!

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