Given a unit vector qer", we saw in Homework 5, Exercise 5 that the nxn matrix Po = qq? orthogonally projects any vector xe R" onto the line spanned by q. Suppose now we have two mutually orthonormal vectors q, q2 Rn. Consider the nxn matrix Pg1.92 = qq? + 2297 a. Show that P91.92 is an orthogonal projector. That is, show P21.92 = P91,92 and P91.92 = P91,92 Note: Just as in the previous homework, P2.., refers to matrix multiplying P0.9 with itself. b. In the subexercises that follow, let q = (1 0 0)*, q2 = (0 ta ta)', and x = (0 1 3)". i. Using GeoGebra, MATLAB, etc. or by hand, sketch Span{q1, 2}. Onto your sketch of Span{q, q2}, plot x and Pg.cx. Connect the tips of these vectors with a straight line. ii. What geometrically does multiplying x by P91,92 do to x? c. Suppose {q, q2, ...,x} forms an orthonormal basis for some k dimensional subspace of Rn. Propose a matrix P91.42.....gk that orthogonally projects any vector xer onto this subspace. d. Suppose {q, q2, ..., k} forms an orthonormal basis for some k dimensional subspace of Rn. Propose a matrix Cg1.92.that orthogonally projects any vector xe R onto the subspace perpendicular to the subspace spanned by {q, q2, ..., qx}. Note: What does it mean for a subspace to be perpendicular to another subspace? It means that any vector in the former subspace is orthogonal to any vector in the latter subspace. For example, in the full QR factorization of a matrix, the subspace spanned by the columns of Q and the subspace spanned by the columns of Q- are perpendicular subspaces. Given a unit vector qer", we saw in Homework 5, Exercise 5 that the nxn matrix Po = qq? orthogonally projects any vector xe R" onto the line spanned by q. Suppose now we have two mutually orthonormal vectors q, q2 Rn. Consider the nxn matrix Pg1.92 = qq? + 2297 a. Show that P91.92 is an orthogonal projector. That is, show P21.92 = P91,92 and P91.92 = P91,92 Note: Just as in the previous homework, P2.., refers to matrix multiplying P0.9 with itself. b. In the subexercises that follow, let q = (1 0 0)*, q2 = (0 ta ta)', and x = (0 1 3)". i. Using GeoGebra, MATLAB, etc. or by hand, sketch Span{q1, 2}. Onto your sketch of Span{q, q2}, plot x and Pg.cx. Connect the tips of these vectors with a straight line. ii. What geometrically does multiplying x by P91,92 do to x? c. Suppose {q, q2, ...,x} forms an orthonormal basis for some k dimensional subspace of Rn. Propose a matrix P91.42.....gk that orthogonally projects any vector xer onto this subspace. d. Suppose {q, q2, ..., k} forms an orthonormal basis for some k dimensional subspace of Rn. Propose a matrix Cg1.92.that orthogonally projects any vector xe R onto the subspace perpendicular to the subspace spanned by {q, q2, ..., qx}. Note: What does it mean for a subspace to be perpendicular to another subspace? It means that any vector in the former subspace is orthogonal to any vector in the latter subspace. For example, in the full QR factorization of a matrix, the subspace spanned by the columns of Q and the subspace spanned by the columns of Q- are perpendicular subspaces
