(Distance between lines) The points P = (x, X, x) and Q = (y, 3 y, -1) are on two lines

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Problems 1-11 use four data points b = (0,8,8,20) to bring out the key ideas.

Project b = (0,8,8,20) onto the line through a = (0,1,3,4). Find x = D and p = xa. The best C in Problems 5-6 and the best D in Problems 7-8 do not agree with the best (C, D) in Problems 1-4. That is because (1, 1, 1, 1) and (0, 1,3,4) are __ perpendicular.

For the closest parabola b = C + Dt + Et 2 to the same four points, write down the unsolvable equations Ax = b in three unknowns x = (C, D, E). Set up the three normal equations AT Ax = ATb (solution not required). In Figure 4.9a you are now fitting a parabola to 4 points-what is happening in Figure 4.9b?

For the closest cubic b = C + Dt + Et 2 + Ft 3 to the same four points, write down the four equations Ax = b. Solve them by elimination. In Figure 4.9a this cubic now goes exactly through the points. What are p and e?

The average of the four times is t = -1-(0 + 1 + 3 + 4) = 2. The average of the four b's is b = -1- (0 + 8 + 8 + 20) = 9. (a) Verify that the best line goes through the center point (t, b) = (2,9). (b) Explain why C + Dt = b comes from the first equation in AT Ax = AT b.

Questions 12-16 introduce basic ideas of statistics-the foundation for least squares.

Questions 12-16 introduce basic ideas of statistics-the foundation for least squares.

Questions 12-16 introduce basic ideas of statistics-the foundation for least squares.

Questions 12-16 introduce basic ideas of statistics-the foundation for least squares.

Questions 12-16 introduce basic ideas of statistics-the foundation for least squares.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

Questions 17-24 give more practice with x and p and e.

What condition on (tI, bd, (t2, b2). (t3, h) puts those three points onto a straight line? A column space answer is: (b l b2 , b3 ) must be a combination of (1, 1, 1) and (tI, t2, t3). Try to reach a specific equation connecting the t's and b's. I should have thought of this question sooner!

Find the plane that gives the best fit to the 4 values b = (0,1,3,4) at the comers (1,0) and (0, 1) and (-1.0) and (0, -1) of a square. The equations C + Dx + Ey = b at those 4 points are Ax = b with 3 unknowns x = (C, D, E). What is A? At the center (0,0) of the square, show that C + D X + E Y = average of the b's.

(Distance between lines) The points P = (x, X, x) and Q = (y, 3 y, -1) are on two lines in space that don't meet. Choose x and y to minimize the squared distance II P - Q 112. The line connecting the closest P and Q is perpendicular to _

Suppose the columns of A are not independent. How could you find a matrix B so that P = B(BT B)-1 BT does give the projection onto the column space of A? (The usual formula will fail when AT A is not invertible.)

Usually there will be exactly one hyperplane in Rn that contains the n given points x = 0, aI, ... ,an-I. (Example for n = 3: There will be one plane containing 0, a I, a2 unless .) What is the test to have exactly one plane in Rn ?