Enter the matrix B and the vector d into MATLAB. To compute projVd, we must first find an orthonormal basis for V$.$ For this, we use the qr$()$ command:

$>>[$Q$,$ R$]=$ qr$($B$,0)$

The columns of Q form an orthonormal basis for V$.$ Let's give them their own names:

$>>$ x $=$ Q$($:$,1)$

$>>$ y $=$ Q$($:$,2)$

Now we're ready to compute projVd. To do this, we add up the projections of d onto each element in our orthonormal basis, like so:

$>>$ v $=$ dot$($x$,$d$)*$x $+$ dot$($y$,$d$)*$y

The resulting vector v here is projVd. Remember to include all your input and output for this procedure in your write$-$up$.$

Now solve the equation Bc $=$ projVd $=$ v by typing in the following:

$>>$ c $=$ B$\backslash$v

Check that your answer is correct by entering

$>>$ B$*$c $-$ v

Make sure the answer is zero. $($Keep in mind that MATLAB may return a very small number instead of zero due to rounding errors.$)$

Now let's compare the answer we just found to what MATLAB gives us when we run the built$-$in least squares algorithm named lscov. To use it$,$ we must specify three parameters: the matrix B$,$ the vector d$,$ and a covariance matrix X$,$ which we won't worry about in this course. Type in the following command:

$>>$ cl $=$ lscov$($B$,$ d$,$ eye$(5))$

How does your answer to this part compare to your previous answer?