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Question is :

Let $v ∈ \mathbb{R^n}$ be a vector with $∥v∥ = 1$.

Consider the mapping $φ_v: \mathbb{R^n} → \mathbb{R^n}$ given by :

$ φ_v (x) = x - 2⟨x, v⟩v.$

Describe the function geometrically.

I'm not sure that i'm understanding the question correctly , how can i describe a mapping geometrically ?

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The question is asking what do you geometrically do to $\vec{x}$ to get $\phi_v\left(\vec{x}\right)$.

For example, if $f\left(\vec{x}\right) = \vec{x} + \vec{1}$, then $f$ geometrically is a translation of $\vec{x}$ by the vector of all ones.

To describe your problem, think of how you compute projections as a hint.

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  • $\begingroup$ thanks for the answer , although i am not sure what does this part means : $2⟨x,v⟩v$. in addition you marked the $x$ as a vector why ? $\endgroup$ – james Nov 14 '17 at 18:56
  • $\begingroup$ @james because $x$ is a vector, i marked it a vector. The notation $<x,v>$ denotes the dot product, i.e. $<x,v> = \vec{x} \cdot \vec{v}$. $\endgroup$ – gt6989b Nov 14 '17 at 19:52
  • $\begingroup$ i think i got it , it's a Vector rejection , right ? $\endgroup$ – james Nov 14 '17 at 20:03

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