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**(a)Let u be a nonzero vector in R[tex]^{n}[/tex]. For all v[tex]\epsilon[/tex]R[tex]^{n}[/tex], show that proj[tex]_{u}[/tex](proj[tex]_{u}[/tex](v)) = proj[tex]_{u}[/tex](v) and proj[tex]_{u}[/tex](v - proj[tex]_{u}[/tex](v)) = [tex]\vec{0}[/tex]**

(b) An alternate proof of the Cauchy-Schwarz inequality. For v,w [tex]\epsilon[/tex]R[tex]^{n}[/tex], consider the function q: R -> R defined by q(t) = ([tex]\vec{v}+t\vec{w}) \bullet (\vec{v}+t\vec{w}).[/tex] Explain why q(t) >= 0 for all t

[tex]\epsilon[/tex]R. By interpreting q(t) as a quadratic polynomial in t, show that |[tex]\vec{v} \bullet \vec{w} <= ||\vec{v}|| ||\vec{w}||.[/tex]

HINT: For a,b,c [tex]\epsilon[/tex]R, we have at^2 +bt + c >= 0 for all t [tex]\epsilon[/tex]R if and only if a > 0 and b^2 - 4ac <= 0.

(b) An alternate proof of the Cauchy-Schwarz inequality. For v,w [tex]\epsilon[/tex]R[tex]^{n}[/tex], consider the function q: R -> R defined by q(t) = ([tex]\vec{v}+t\vec{w}) \bullet (\vec{v}+t\vec{w}).[/tex] Explain why q(t) >= 0 for all t

[tex]\epsilon[/tex]R. By interpreting q(t) as a quadratic polynomial in t, show that |[tex]\vec{v} \bullet \vec{w} <= ||\vec{v}|| ||\vec{w}||.[/tex]

HINT: For a,b,c [tex]\epsilon[/tex]R, we have at^2 +bt + c >= 0 for all t [tex]\epsilon[/tex]R if and only if a > 0 and b^2 - 4ac <= 0.

um i don't know how to fix the formula but i think you understand what I mean.

I really need a starting point, i have no idea how to do it, especially part (b), for part a i have tried proving lhs = rhs by starting with lhs, but nothing is cancelling. I just used the projection formula and projected v onto u and then nothing cancels so i project the (projected v onto u) piece onto the u again, and i am having a hard time simplifying, i'm not sure if this is even the right way to solve it.

any information would be great