Lecture 5: Column spaces and QR (part II) | Highlights and Annotations by Gistr.

The lecture describes projecting a vector onto a span of orthonormal vectors, showing the projection is a sum of individual projections. Gram-Schmidt orthogonalization is introduced, creating an orthonormal basis from a set of linearly independent vectors. However, Gram-Schmidt's numerical instability is highlighted, leading to the introduction of Householder QR factorization, a more stable alternative using reflection matrices to achieve QR decomposition. This segment explains the formula for projecting a vector onto a single unit vector and extends it to projecting onto the span of multiple vectors, introducing the concept of coefficients (C1 to CK) for the projection. This segment details the process of finding the projection of vector B onto the span of vectors A1 to AK by minimizing the squared error. It sets up the equation involving a double sum and dot products, laying the groundwork for further simplification. This segment simplifies the double sum equation from the previous segment using the orthogonality of the vectors. It leverages the properties of orthonormal vectors to reduce the equation, making it easier to find the minimum error.This segment shows how to minimize the error equation derived earlier by taking the derivative with respect to each coefficient and setting it to zero. The result demonstrates that the projection onto the span of orthonormal vectors is a simple sum of individual projections. This segment introduces the Gram-Schmidt process, a method for orthogonalizing a set of vectors. It explains the basic idea of the algorithm and sets the stage for a more detailed explanation in the following segments. This segment illustrates the numerical instability of the Gram-Schmidt process using a simple example. It highlights the problem of dividing by very small numbers, which can lead to significant errors and unreliable results. This segment delves into the crucial step of determining the value of 'C' within the Householder reflection method used for QR factorization. The speaker explains how a specific choice for 'B' (based on the parallelism of vectors) implicitly defines 'C', resolving a system with initially two unknowns. The explanation connects the mathematical manipulation with the broader goal of the QR factorization process, highlighting the elegance and efficiency of the Householder method.