# ✅ Checkpoint: Matrices and Transformations

By Better Wise



## Quiz

1. 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
2. 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
3. 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)
4. 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
5. 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)
6. For a 2x2 matrix [[a, b], [c, d]], the determinant is ____.
7. The rank of an m by n matrix can never exceed ____.
8. To convert standard coordinates v into basis-B coordinates, you solve B c = v, which in NumPy is written np.linalg.____(B, v).
9. 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.
10. 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.


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## Contents

Part of [🧮 Mathematics for Machine Learning 1: Linear Algebra Foundations](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations.md)

- **📐 Vectors, Norms and Similarity**
  - [➡️ Vectors as Arrows, Points, and NumPy Arrays](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/vectors-as-arrows-points-and-numpy-arrays.md)
  - [📏 Norms and Distances: Measuring Vector Size](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/norms-and-distances-measuring-vector-size.md) (sign-in required)
  - [🎯 Dot Product and Cosine Similarity](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/dot-product-and-cosine-similarity.md)
  - [✅ Checkpoint: Vectors and Similarity](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/checkpoint-vectors-and-similarity.md)
- **🔄 Matrices and Transformations**
  - [🧱 Matrices as Transformations and Matrix Multiplication as Composition](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/matrices-as-transformations-and-matrix-composition.md)
  - [🧭 Inverse, Rank, and Span](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/inverse-rank-and-span.md)
  - [🔀 Basis and Change of Basis](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/basis-and-change-of-basis.md)
  - [✅ Checkpoint: Matrices and Transformations](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/checkpoint-matrices-and-transformations.md) — this lesson
- **🔆 Projections, Eigenvectors and SVD**
  - [📉 Projections and Least Squares Intuition](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/projections-and-least-squares-intuition.md)
  - [🌀 Eigenvectors and Eigen-Decomposition](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/eigenvectors-and-eigen-decomposition.md)
  - [🔍 A First Look at SVD](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/first-look-at-svd.md) (premium content)
  - [✅ Checkpoint: Projections, Eigen and SVD](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/checkpoint-projections-eigen-and-svd.md) (premium content)
- [Course cheatsheet](https://everwise.study/courses/mathematics-for-machine-learning-1-linear-algebra-foundations/cheatsheet.md) (premium content)