Exercises
Use these exercises to check the chapter ideas. Work by hand first.
Solve:
What is x?
Compute it first, then check your number.
Let v_1 = [1, 2] and v_2 = [3, 0].
Compute 2v_1 + v_2.
Compute it first, then check your number.
The vectors [1, 0], [0, 1], and [1, 1] span the plane.
Do they form a basis for the plane?
Answer it first, then check.
The matrix
has columns [2, 0] and [4, 0]. What is its rank?
Compute it first, then check your number.
For the same matrix, is [6, 0] in the column space?
Answer it first, then check.
The same matrix maps [x, y] to [2x + 4y, 0].
Does [-2, 1] map to zero?
Answer it first, then check.
Let:
What is the eigenvalue for eigenvector [1, 0]?
Compute it first, then check your number.
In A = PDP^{-1}, which factor acts first on an input vector?
Enter P, D, or P^{-1}.
Answer it first, then check.
If singular values are 8, 3, 0.5, and 0.1, which two values does a
rank-2 approximation keep?
Compute it first, then check your number.
Does the first principal component always represent the most causal factor?
Answer it first, then check.
A matrix has one solution x_0 to Ax = b. It also has a nonzero vector z
with Az = 0.
Enter 1 if x_0 + z is also a solution to Ax = b.
Compute it first, then check your number.
Hint
Use distributivity: A(x_0 + z) = Ax_0 + Az.
Solution
Since Ax_0 = b and Az = 0:
So x_0 + z is also a solution.
This is the practical meaning of a null direction: it can be added to an input without changing the output. That is why one solution can generate another.
A 3 x 3 matrix has rank 2.
Enter 1 if at least one independent input direction is lost or collapsed by
the matrix.
Compute it first, then check your number.
Hint
The matrix starts with three input coordinates but only preserves two independent output directions.
Solution
Enter 1. Rank 2 means the output uses two independent directions. From a
three-dimensional input, at least one independent direction is collapsed or made
redundant in the output.
In SVD, singular values describe strengths of directions.
Enter 1 if a singular value of 0 means that direction is collapsed.
Compute it first, then check your number.
Hint
Singular values are nonnegative strengths.
Solution
Enter 1. A singular value of 0 means the matrix does not carry that
direction into a nonzero output.
A rank-2 approximation keeps the two largest singular values.
Enter 1 if this can still remove small details that matter for a specific
task.
Compute it first, then check your number.
Hint
Ask whether weak directions can contain rare but important signal.
Solution
Enter 1. Low-rank approximation keeps the strongest directions, but a weaker
direction can still carry task-relevant detail.
Next
Use the hints only after you have tried the exercises. Use the solutions after you can explain where you got stuck.