Exercises
Exercise 1
The aim of this exercise is to establish a basic intuition of metrics and sets as well as some fundamental proof techniques.
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Exercise 2
The aim of this exercise is to foster an intuition for matrix algebra and index notation as well as refresh knowledge about systems of linear equations and how to solve them.
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Exercise 3
The aim of this exercise is to establish an understanding of vector spaces and their connection to systems of linear equations and linear maps.
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Exercise 4
The aim of this exercise is to practice calculating determinants, eigen-stuff and matrix decompositions, as well as strengthen the intuition behind these concepts.
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Exercise 5
The aim of this exercise is to deepen the understanding of inner products and orthogonal projections.
Helpful Additional Resources
- Trig Cheat sheet with unit circle, radian to degree, etc.
- A big application of orthogonal projections that we did not get around to this week is Least Squares. I encourage you to read up on this in Chapter 6.5 LEAST-SQUARES PROBLEMS of Linear Algebra and Its Applications (Lay) and/or on this page of "Understanding Linear Algebra".
Exercise 6
The aim of this exercise is to promote the understanding of sequences, series, and limits and formally establish the concept of univariate differentiation.
Helpful Additional Resources
- If you want to deepen your understanding of this week's topics and related concepts, I recommend chapters 7-19 of this MIT open course ware course.
Exercise 7
The aim of this exercise is to practice matrix calculus, specifically computing derivatives of functions from and to spaces of different dimensions.
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Exercise 8
The aim of this exercise is to become familiar with the Hessian matrix and local extrema and foster an inutition for Taylor series.
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Exercise 9
The aim of this exercise is to become familiar with the concepts of Convexity and Lipschitz continuity and provide an intuition for their use for optimization problems.
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Exercise 10
The aim of this exercise is to introduce probability spaces and measure integrals, which form a fundamental basis for the curriculum of the next two weeks.
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Exercise 11
The aim of this exercise is to establish the concepts of random variables and distributions form a probability-theory perspective.
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Exercise 12
The aim of this exercise is to foster an understanding of conditional expectation and conditional variance.