40.5: System Properties

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The above methods for solving systems of linear equations is only part of the story. We also explored ways to understand properties of linear systems. Properties such as rank, determinate, eigenvectors and eigenvalues all provide insight into the matrices that are at the core of the systems.

One problem is that as systems get really large the computational cost of finding a solution can also become large and intractable (i.e. difficult to solve). We use our understanding of matrix properties and “decompositions” to transform systems into simpler forms so that solving the problem also becomes simple.

In class tomorrow we will review all of these concepts by looking at methods to solve linear systmes of the form \(Ax=b\) using \(QR\) decomposition. When we solve for \(Ax=b\) with QR decomposition. We have the following steps:

Find the \(QR\) decomposition of \(A\) such that:
- \(R\) is square upper-triangular matrix
- The Columns of \(Q\) are orthonormal
From \(QRx=b\), we obtain \(Rx = Q^{\top}b\)
Solve for \(x\) using back substitution.

Do This

Search for a video describing the \(QR\) decomposition of a matrix. Try to pick a video that you think does a good job in a short amount of time.