From No Bullshit Guide to Linear Algebra by Ivan Savov.
Fundamentals
Exponentiation
- \(a^{-b} = 1 / a^b\)
- \(a^{1/2} = \sqrt{a} = \sqrt[2]{a}\)
- \(a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}\)
Function inverses
| function \(f(x)\) | inverse \(f^{-1}(x)\) |
|---|---|
| \(x\) | \(-x\) |
| \(x + 2\) | \(x - 2\) |
| \(2x\) | \(\frac{1}{2}x\) |
| \(3x + 5\) | \(\frac{1}{3}(x - 5)\) |
| \(x^2\) | \(\pm\sqrt{x}\) |
| \(2^x\) | \(\log_2(x)\) |
| \(a^x\) | \(\log_a(x)\) |
| \(exp(x) \equiv e^x\) | \(ln(x) \equiv \log_e(x)\) |
| \(sin(x)\) | \(sin^{-1}(x) \equiv \arcsin(x)\) |
| \(cos(x)\) | \(cos^{-1}(x) \equiv \arccos(x)\) |
Example:
- \(\log_5(3 + \sqrt{6 \sqrt{x} - 7}) = 42 \theta\)
- \(x = (\frac{1}{6}((5^{42 \ theta} - 3)^2 + 7))^2\)
Factoring
- \(6x^2y + 15x = (3)(2)(x)(x)y + (5)(3)x = 3x(2xy + 5)\)