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@ -40,11 +40,11 @@ useful study guide.
## Lecture 2 (Feb 8)
* Matrix multiplication by blocks and columns-times-rows. Complexity: standard algorithm for (m×p)⋅(p×n) is [Θ(mnp): roughly proportional](https://en.wikipedia.org/wiki/Big_O_notation) to mnp for large m,n,p. (There also [exist theoretically better](https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication), but highly impractical, algorithms.)
* Briefly reviewed the "famous four" matrix factorizations: [LU](https://en.wikipedia.org/wiki/LU_decomposition), [diagonalization XΛX⁻¹ or QΛQᵀ](https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix), [QR](https://en.wikipedia.org/wiki/QR_decomposition), and the [SVD UΣVᵀ](https://en.wikipedia.org/wiki/Singular_value_decomposition).
* Matrix multiplication by blocks and columns-times-rows. Complexity: standard algorithm for (m×p)⋅(p×n) is [Θ(mnp): roughly proportional](https://en.wikipedia.org/wiki/Big_O_notation) to mnp for large m,n,p, regardless of how we rearrange it into blocks. (There also [exist theoretically better](https://en.wikipedia.org/wiki/Computational_complexity_of_matrix_multiplication), but highly impractical, algorithms.)
* Briefly reviewed the "famous four" matrix factorizations: [LU](https://en.wikipedia.org/wiki/LU_decomposition), [diagonalization XΛX⁻¹ or QΛQᵀ](https://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix), [QR](https://en.wikipedia.org/wiki/QR_decomposition), and the [SVD UΣVᵀ](https://en.wikipedia.org/wiki/Singular_value_decomposition). QR and QΛQᵀ in the columns-times-rows picture, especially QΛQᵀ (diagonalization for real A=Aᵀ) as a sum of symmetric rank-1 projections.
* The [four fundamental subspaces](https://web.mit.edu/18.06/www/Essays/newpaper_ver3.pdf) for an m×n matrix A of rank r, mapping "inputs" x∈ℝⁿ to "outputs" Ax∈ℝᵐ: the "input" subspaces C(Aᵀ) (row space, dimension r) and its [orthogonal complement](https://en.wikipedia.org/wiki/Orthogonal_complement) N(A) (nullspace, dimension n–r); and the "output" subspaces C(A) (column space, dimension r) and its orthogonal complement N(Aᵀ) (left nullspace, dimension m–r).
**Further reading**: Textbook 1.3–1.6. [OCW lecture 2](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-2-multiplying-and-factoring-matrices/). If you haven't seen [matrix multiplication by blocks](https://www.math.ucdavis.edu/~linear/old/notes9.pdf) before, [here is a nice video](https://www.youtube.com/watch?v=KCUgWj5nhYc)
**Further reading**: Textbook 1.3–1.6. [OCW lecture 2](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-2-multiplying-and-factoring-matrices/). If you haven't seen [matrix multiplication by blocks](https://www.math.ucdavis.edu/~linear/old/notes9.pdf) before, [here is a nice video](https://www.youtube.com/watch?v=KCUgWj5nhYc).
## *Optional* Julia Tutorial: Wed Feb 8 @ 5pm [via Zoom](https://mit.zoom.us/j/96829722642?pwd=TDhhME0wbmx0SG5RcnFOS3VScTA5Zz09)