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Problem 1. We say that P is a projection matrix if P^T = P and P^2 = P. (a) If P is a projection, show that I - P is also a projection. (b) Show that the projections P and I - P satisfy P(I - P) = 0. (c) Let A be any matrix of shape m x n so that A^T A is square of shape n x n. Assuming that the inverse (A^T A)^-1 exists, show that P = A(A^T A)^-1 A^T is a projection matrix. [We saw in class that this matrix projects onto the column space of A.] (d) In the special case that A is a square and invertible, show that P = A(A^T A)^-1 A^T = I. What does this mean? Problem 2. Consider the plane x + 2y + 2z = 0 with normal vector a = (1, 2, 2). (a) Use the formula from 1(c) to find the 3 x 3 matrix P that projects onto the line ta. [Hint: Just let A = a.] (b) Use the matrix P to project the vector b = (1, -1, 1) onto the line. (c) Find two vectors in the plane x + 2y + 2z = 0 and then use the formula from 1(c) to find the 3 x 3 matrix Q that projects onto the plane. [Hint: Let A be the 3 x 2 matrix whose columns are the two vectors that you found.] (d) Use the matrix Q to project the vector b = (1, -1, 1) onto the plane. (e) Finally, check that P + Q = I. Does this surprise you? Problem 3. Shortcut. Let a = (1, 2, -1, 1) and consider the following hyperplane in R^4: a^T x = 1x_1 + 2x_2 - 1x_3 + 1x_4 = 0. (a) Use 1(c) to compute the matrix P that projects onto the line ta. (b) We could also use 1(c) to compute the matrix Q that projects onto the hyperplane, but this would take too long. Instead, use the shortcut formula Q = I - P. (c) Project the point (1, 2, 3, 4) onto the hyperplane.

          Problem 1. We say that P is a projection matrix if P^T = P and P^2 = P.
(a) If P is a projection, show that I - P is also a projection.
(b) Show that the projections P and I - P satisfy P(I - P) = 0.
(c) Let A be any matrix of shape m x n so that A^T A is square of shape n x n. Assuming
that the inverse (A^T A)^-1 exists, show that P = A(A^T A)^-1 A^T is a projection matrix.
[We saw in class that this matrix projects onto the column space of A.]
(d) In the special case that A is a square and invertible, show that P = A(A^T A)^-1 A^T = I.
What does this mean?

Problem 2. Consider the plane x + 2y + 2z = 0 with normal vector a = (1, 2, 2).
(a) Use the formula from 1(c) to find the 3 x 3 matrix P that projects onto the line ta.
[Hint: Just let A = a.]
(b) Use the matrix P to project the vector b = (1, -1, 1) onto the line.
(c) Find two vectors in the plane x + 2y + 2z = 0 and then use the formula from 1(c) to
find the 3 x 3 matrix Q that projects onto the plane. [Hint: Let A be the 3 x 2 matrix
whose columns are the two vectors that you found.]
(d) Use the matrix Q to project the vector b = (1, -1, 1) onto the plane.
(e) Finally, check that P + Q = I. Does this surprise you?

Problem 3. Shortcut. Let a = (1, 2, -1, 1) and consider the following hyperplane in R^4:
a^T x = 1x_1 + 2x_2 - 1x_3 + 1x_4 = 0.
(a) Use 1(c) to compute the matrix P that projects onto the line ta.
(b) We could also use 1(c) to compute the matrix Q that projects onto the hyperplane,
but this would take too long. Instead, use the shortcut formula Q = I - P.
(c) Project the point (1, 2, 3, 4) onto the hyperplane.

Problem 1. We say that P is a projection matrix if P^T = P and P^2 = P.
(a) If P is a projection, show that I - P is also a projection.
(b) Show that the projections P and I - P satisfy P(I - P) = 0.
(c) Let A be any matrix of shape m x n so that A^T A is square of shape n x n. Assuming
that the inverse (A^T A)^-1 exists, show that P = A(A^T A)^-1 A^T is a projection matrix.
[We saw in class that this matrix projects onto the column space of A.]
(d) In the special case that A is a square and invertible, show that P = A(A^T A)^-1 A^T = I.
What does this mean?

Problem 2. Consider the plane x + 2y + 2z = 0 with normal vector a = (1, 2, 2).
(a) Use the formula from 1(c) to find the 3 x 3 matrix P that projects onto the line ta.
[Hint: Just let A = a.]
(b) Use the matrix P to project the vector b = (1, -1, 1) onto the line.
(c) Find two vectors in the plane x + 2y + 2z = 0 and then use the formula from 1(c) to
find the 3 x 3 matrix Q that projects onto the plane. [Hint: Let A be the 3 x 2 matrix
whose columns are the two vectors that you found.]
(d) Use the matrix Q to project the vector b = (1, -1, 1) onto the plane.
(e) Finally, check that P + Q = I. Does this surprise you?

Problem 3. Shortcut. Let a = (1, 2, -1, 1) and consider the following hyperplane in R^4:
a^T x = 1x1 + 2x2 - 1x3 + 1x4 = 0.
(a) Use 1(c) to compute the matrix P that projects onto the line ta.
(b) We could also use 1(c) to compute the matrix Q that projects onto the hyperplane,
but this would take too long. Instead, use the shortcut formula Q = I - P.
(c) Project the point (1, 2, 3, 4) onto the hyperplane.

Added by Frank T.

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