For the least squares line y = a + ct fitted to (0,0), (1,2), (2,1), the normal equations are 3a + 3c = 3 and 3a + 5c = 4. Enter the slope c as a decimal.

The projection of b = (2,0) onto the line spanned by u = (1,1) is (1,1). What is its projection onto the line spanned by 2u = (2,2)?

• (2,2), because doubling a direction doubles every projection.
• (1,1), because both directions span the same line.
• (0.5,0.5), because doubling a direction halves every projection.
• (2,0), because a longer direction makes the residual vanish.

Let A = [[1,1],[1,1]] and b = (1,3). Both normal equations are 2x1 + 2x2 = 4. What follows about x = (0,2) and x = (1,1)?

• Both minimise the squared residual and give the same fitted vector.
• Only (0,2) minimises the squared residual because one coefficient is zero.
• Only (1,1) minimises the squared residual because its coefficients agree.
• Neither minimises the squared residual because neither fits b exactly.

Using the standard dot product, b = (5,1) has projection p = (3,3) onto the line spanned by u = (1,1). Which gives r = b - p and r dot u?

• (2, 2); dot product 4
• (-2, 2); dot product 0
• (0, 0); dot product 0
• (2, -2); dot product 0

Using the standard dot product, project (3, 7) onto the line spanned by (1, 1). The projection is k(1, 1). Enter k.

A fitted line has vertical residuals 3 and -3. Why does adding these signed residuals fail to measure its total error?

• Their sum makes both observed points lie on the fitted line.
• Their sum measures the perpendicular distance to the fitted line.
• Their sum is zero even though both observations have nonzero errors.

Let p be the projection of b onto the column space and r = b - p. Why does any x with Ax = p minimise the squared residual length?

• The residual becomes zero for every possible data vector b.
• Every other fitted vector adds a positive squared distance.
• Every column becomes longer than every possible residual vector.

For a real matrix A, Ax = b is inconsistent. What does any solution x of A^T A x = A^T b produce?

• Ax equals b despite the stated inconsistency.
• Ax has the largest possible distance from b.
• Ax equals the projection of b onto the column space.

Using exact arithmetic and the standard dot product, a student claims x solves A^T A x = A^T b. The computed residual b - Ax has a nonzero dot product with a column of A. What follows?

• The claimed solution or the residual calculation contains an error.
• The normal equations require nonzero residual dots for rectangular matrices.
• The nonzero dot proves that the original system has no least squares solution.

Fit y = a + ct by least squares to (0,1), (1,2), (2,2), (3,3), using vertical residuals. Which line results from the normal equations?

• y = 0.6 + 1.1t
• y = 1.1 + 0.6t
• y = 1.0 + 0.7t