Checkpoint: Matrices and Transformations

By Better Wise

Quiz

For the matrix M = [[1, 1], [0, 1]], where does the basis vector e2 = (0, 1) land?
- (2, 1), because the unit square corner (1, 1) moves there
- (1, 1), because the second column of M is (1, 1)
- (1, 0), because the first column of M is (1, 0)
- (0, 1), because e2 stays fixed
With A = [[0, -1], [1, 0]] (rotate 90 degrees) and B = [[1, 1], [0, 1]] (shear), what does A @ B mean as a transformation?
- Apply both at the same time, so order does not matter
- Apply A first, then B
- Apply B first, then A
- Apply B twice, then A once
A 2x2 matrix has columns (1, 2) and (2, 4). What are its rank and the span of its columns?
- Rank 1, span is the line through (1, 2)
- Rank 2, span is the whole plane
- Rank 0, span is a single point
- Rank 1, span is the line through (1, 0)
What does a determinant of zero tell you about a square matrix?
- The transformation preserves area and orientation
- Space collapsed to a lower dimension, so the matrix is singular and has no inverse
- The matrix is invertible but numerically unstable
- The matrix is the identity
With basis b1 = (1, 1) and b2 = (-1, 1), what are the coordinates of v = (3, 1) in this basis?
- (2, -1), since 2*b1 - 1*b2 = (3, 1)
- (-1, 2), because you swap the components
- (3, 1), because coordinates do not change
- (1, 1), since b1 + b2 = (0, 2)
For a 2x2 matrix [[a, b], [c, d]], the determinant is ____.
The rank of an m by n matrix can never exceed ____.
To convert standard coordinates v into basis-B coordinates, you solve B c = v, which in NumPy is written np.linalg.____(B, v).
In your own words, explain why matrix multiplication is not commutative, using the rotate-then-shear example. Then describe how you would check in NumPy that A @ B differs from B @ A.
A classmate claims that the vectors (1, 1) and (2, 2) form a basis of R^2. Explain why they do not, and describe how you would test a candidate set of vectors for being a basis in NumPy.