Least squares approximation

[florence]

1. Let x, y ∈ Rn be two non-zero vectors. The least squares approximation of y in span {x} is the vector cx, with c ∈ R, that minimises the expression ∥y − cx∥^2.

1. Find an expression for the minimiser cˆ in terms of x and y.
2. Express the approximation error ∥y − cˆx∥ in terms of y and the angle θ between x and y. As
   always, simplify your answer as far as possible.

[work attached in response #3 ]
Can somebody please help me out on this question?

 

[Deleted member 4993]

1. Let x, y ∈ Rn be two non-zero vectors. The least squares approximation of y in span {x} is the vector cx, with c ∈ R, that minimises the expression ∥y − cx∥^2.

1. Find an expression for the minimiser cˆ in terms of x and y.
2. Express the approximation error ∥y − cˆx∥ in terms of y and the angle θ between x and y. As
   always, simplify your answer as far as possible.

Can somebody please help me out on this question?

Please show us what you have tried and exactly where you are stuck.

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[florence]

1. Let x, y ∈ Rn be two non-zero vectors. The least squares approximation of y in span {x} is the vector cx, with c ∈ R, that minimises the expression ∥y − cx∥^2.

1. Find an expression for the minimiser cˆ in terms of x and y.
2. Express the approximation error ∥y − cˆx∥ in terms of y and the angle θ between x and y. As
   always, simplify your answer as far as possible.

This is my attempt but I don’t feel confident about the answer