Linear algebra - orthogonal projections & inner products?

[rsingh628]

Hello all, I am having difficulties setting up the problem below.



Seems to me that the vector closest to x from the subspace spanned by T will be the vector in T which is orthogonal to x. What I am stuck with is using the inner product definition to set-up the Gram matrix. What would be the correct approach? Appreciate any feedback.

 

[lex]

I would just choose a general element in T - say [MATH]\hspace1ex \boldsymbol{y}=\binom{\lambda}{2\lambda}[/MATH]Write down an expression for [MATH]||\boldsymbol{y}-\boldsymbol{x}||[/MATH]and minimise that expression.
I got [MATH]\widehat{\boldsymbol{x}}=\begin{pmatrix} \frac{5}{14} \\ \frac{5}{7} \end{pmatrix}[/MATH]