Math/Linear algebra
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SVD, matrix inverse, and pseudoinverseMath/Linear algebra 2020. 1. 25. 10:36
1. Overview 2. Description 2.1 Inverse Full rank square matrix A Now I'm going to invert A which is fine we assume for the moment that A is an invertible matrix. So it's square and full rank. And of course, whatever operation you perform on one side of the equation must be repeated on the other side of the equation. So we apply the inverse to the right-hand side as well. Now we know that each of..
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Least-squares for model fittingMath/Linear algebra 2020. 1. 24. 14:20
1. Overview There are uncountable dynamics and processes and individuals with uncountable and mind-bogglingly complex interactions. So how can we possibly make sense of all of this complexity? The answer is we can't :( So instead we generate simple models and we fit models to data using linear least squares modeling and that is the goal of this section of the course. the idea of building models ..
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Linear Algebra FeaturesMath/Linear algebra 2020. 1. 23. 18:03
1. Overview Summarize Terminologies and features 2. Description 2.1 Matrix multiplications It doesn't matter if you're multiplying $A^{T}A$ or $AA^{T}$ both results will produce not only a Square matrix but a Symmetric matrix. 2.1.1 Characteristic equation The characteristic equation is the equation that is solved to find a matrix's eigenvalues, also called the characteristic polynomial. For a g..
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Quadratic formMath/Linear algebra 2019. 10. 13. 21:21
1. Overview all matrix is square 2. Description 3. Example What do they mean and how do you interpret them? So this is related to the concept of the energy in matrix S that I referred to. If symmetric 3 by 3 matrix identity matrix 4. Quadratic form in geometry You should also note that all of these surfaces have a value of zero at the coordinate point W1 W2 both equals 0. And that's because when..
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Condition numberMath/Linear algebra 2019. 10. 13. 21:04
1. Overview 2. Description In fact, one way to interpret that condition number is as an indicator of the dominance of large scale structure in the matrix. now the condition number is really small is a well-conditioned matrix but this matrix doesn't really contain any useful information. It doesn't actually contain any meaningful patterns it's literally just noise. In this video, 3. Example 4. Re..
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Projections in RN spaceMath/Linear algebra 2019. 10. 12. 11:59
1. Overview A projection is a linear transformation P from a vector space to itself such that$P^{2}=P$. That is, whenever P is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of ..
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Left inverse and right inverseMath/Linear algebra 2019. 10. 12. 10:08
1. Overview 2. Description 2.1 Tall matrix and left inverse Now the thing is that this matrix definitely does not have a full inverse. However, with a little bit of creativity, we can come up with a way to multiply this matrix to produce the identity matrix. However, what we are inverting is the product A transpose A. So it turns out that A transpose A inverse times A transpose A is going to pro..