Exercise 20.11. Consider the plane P= span Let x- e (a) Calculate Projp(x) by using the Gram-Schmidt process to find an orthogonal basis for P. (b) Calculate Projp(x) by finding a (nonzero) normal vector to Pand then subtracting the projection onto that. (c) By applying the method from (b) to the other vectors e, and ey in the standard basis for R°, compute the 3 x 3 matrix A representing Projp : R → R'. (d) Calculate Projp(x) by using the self-contained recipe in Theorem20.5.3dllustrated in Example Theorem 20.5.3. For linearly independent v., . Vm € R" spanning a subspace W of R" and the associated n x m matrix V whose jth column is vj. the projection of any x e R" into W is Projw x = cvi + eva +..+ CVm where the coeficients es,..,cm are the entries in the m- - (V'V)-'V"x. vector

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A: The complete solutions are given below