Foundations — NumPy linear algebra — np.linalg.solve, eig, svd, norm, det

Before you can read the parent note, you must own every squiggle it uses. This page takes each symbol, gives it a plain-words meaning, ties it to a picture, and says why the topic can't live without it. Read top to bottom—each one leans on the one above.

1. A number and a list of numbers

The little symbol ${\mathbb{R}}^n$ means "a list of $n$ real numbers." So ${\mathbb{R}}^2$ is arrows in a flat plane (two numbers each), ${\mathbb{R}}^3$ is arrows in space. Read $x\in {\mathbb{R}}^n$ as "$x$ is one arrow living in $n$-dimensional space." The symbol $\in$ just means "is a member of / lives in."

2. Subscripts — naming the pieces

The picture: a vector is a stack of boxes; the subscript is the box's floor number. This tiny notation is how the parent note writes $\Vert x{\Vert }_2=\sqrt{{\sum }_i{x}_i^2}$ — it needs to talk about "each component in turn."

3. The matrix — a grid of numbers

4. Matrix times vector — running the machine

The picture: you feed the arrow $\left(\begin{array}{c}x\\ y\end{array}\right)$ into the machine $A$ and it spits out a new arrow. Equivalently: $Ax=x\cdot (\text{column 1})+y\cdot (\text{column 2})$ — a stretched-and-added mix of the two landing arrows from §3.

That example is exactly the parent note's system $Ax=b$ running forwards. Solving is running it backwards.

5. The identity and the inverse

The picture: if $A$ rotates an arrow $30^{\circ }$ and stretches it double, then $A^{-1}$ rotates $30^{\circ }$ back and halves it. Some machines can't be undone—if $A$ flattens the whole plane onto a line, information is lost forever and no $A^{-1}$ exists. That "can't undo" case is exactly what the determinant will detect.

6. The transpose $A^{\top }$

The picture: hold the top-left corner fixed and reflect the grid across the main diagonal (top-left to bottom-right). The parent uses $A^{\top }A$ in the SVD section and $V^{\top }$ in $A=U\Sigma V^{\top }$, so you must recognise this flip.

7. The square root and the sum sign

The picture: these two together build length. The parent's $\Vert x{\Vert }_2=\sqrt{{\sum }_i{x}_i^2}$ is nothing but the Pythagorean theorem—the straight-line distance from the origin to the tip of the arrow. The double bars $\Vert \cdot \Vert$ are read "the size / length of." See Vector norms and metric spaces for the full menu of "size."

8. The Greek letters the parent throws at you

9. Determinant, singular, orthogonal — the vocabulary

The picture for the determinant: feed the unit square (corners at the two basic arrows) into $A$; it comes out as a parallelogram. The determinant is that parallelogram's signed area. If the parallelogram has collapsed to a flat line, its area is $0$—that's the singular case.

How the foundations feed the topic

Equipment checklist