Linear Algebra

Diophantine Equations and the Extended Euclidean Algorithm

Diophantine equations are linear equations with integer solutions.

I’m surprised by how many mathematical theories have been invented out of the need of solving equations or finding the roots of polynomials.

Introduction to Complex Numbers: Lecture 1

• Complex Numbers and Polynomials
  • Motivation for complex numbers: They arise from the need to find roots of polynomials that do not have real solutions.

When your linear algebra course boils down to one theorem https://x.com/Anthony_Bonato/status/1880634250065748471

THEOREM 5.1.5 Equivalent Statements

If $A$ is an $n\times n$ matrix, then the following statements are equivalent: (a) $A$ is invertible. (b) $A\mathbf{x}=0$ has only the trivial solution. (c) The reduced row echelon form of $A$ is $I_n$. (d) A is expressible as a product of elementary matrices. (e) $A\mathbf{x}=\mathbf{b}$ is consistent for every $n\times 1$ matrix $\mathbf{b}$. ( $f$ ) Ax $=\mathbf{b}$ has exactly one solution for every $n\times 1$ matrix $\mathbf{b}$. (g) $\det (A)\ne 0$. (h) The column vectors of $A$ are linearly independent. (i) The row vectors of A are linearly independent. ( $j$ ) The column vectors of $A$ span $R^n$. (k) The row vectors of $A$ span $R^n$. (l) The column vectors of A form a basis for $R^n$. (m) The row vectors of $A$ form a basis for $R^n$. ( $n$ ) A has rank $n$. (o) A has nullity 0 . (p) The orthogonal complement of the null space of $A$ is $R^n$. (q) The orthogonal complement of the row space of $A$ is $\{0\}$. ( $r$ ) The kernel of $T_A$ is $\{0\}$. (s) The range of $T_A$ is $R^n$. (t) $T_A$ is one-to-one. (u) $\lambda =0$ is not an eigenvalue of $A$.