Linear Algebra
Going through Gilbert Strang’s Linear Algebra MIT course.
Lecture 1: The geometry of linear equations
Linear equations as matrices
\[ 2x - y = 0 \\ -x + 2y = 3 \\ \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \\ Ax = b \]
Row picture
- Each row in \(Ax = b\) is a line.
- Intersection of the lines is the solution.
Column picture
Instead of \[ \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \\ \] we think of \[ x \begin{bmatrix} 2 \\ -1 \end{bmatrix} + y \begin{bmatrix} -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \]
We want to find a linear weighting of \([2, -1]^T\) and \([-1, 2]^T\) that reaches \([0, 3]^T\).
Lecture 2: Elimination with Matrices
Identity matrix
\[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \]
Permutation matrix
\[ P_{1,2} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \] #### Rows v. columns - \(P_{1,2}A\) will swap the 1st and 2nd rows of \(A\). - \(AP_{1,2}\) will swap the 1st and 2nd columns of \(A\).
Elimination algorithm
Give an \(n \times n\) matrix:
- Take the first pivot \((1, 1)\).
- Set \((1, 2), ..., (1, n)\) to zero by subtracting the first row.
- Repeat with the next pivots \((2, 2), ..., (n, n)\)
The resulting matrix is the upper triangular matrix \(U\).
Elimination matrices
We can define the subtractions in the elimination algorithm with elimination matrices.
For example, to subtract 4 row 1s from 2, we use: \[ E_{1,2} = \begin{bmatrix} 1 & 0 & 0 \\ -4 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \]
Thus, the elimination algorithm is repeating this step: \[ E_{2,3}(E_{1,3}(E_{1,2}A)) = U \\ EA = U \]