📖 Random Forest Like bagging, but at each split only a random subset of $m \\approx \\sqrt{p}$ predictors is considered. This decorrelates trees, reducing variance further. From: Intro to Statistical Learning Learn more:

Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 The German Path Germany did not abandon liberalism because it failed, but because Germany never fully embraced it. What failed was socialism—but socialists blamed liberalism for problems socialism created. From: The Road to Serfdom Learn more:

The Road to Serfdom | Magic Internet Math

An interactive course exploring F.A. Hayek

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Natural Isomorphism to Double Dual The map $\\phi: V \\to V^{**}$ defined by $\\phi(v)(f) = f(v)$ is a natural isomorphism for finite-dimensional $V$. Proof: **Linearity:** For $u, v \\in V$ and $f \\in V^*$: $$\\phi(u + v)(f) = f(u + v) = f(u) + f(v) = \\phi(u)(f) + \\phi(v)(f)$$ **Injectivity:** If $\\phi(v) = 0$, then $f(v) = 0$ for all $f \\in V^*$. Choosing $f$ from the dual basis shows $v = \\mathbf{0}$. **Dimension:** $\\dim(V) = \\dim(V^*) =... From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Intersection of Subspaces The intersection of any collection of subspaces of $V$ is a subspace of $V$. Proof: Let $\\{W_i\\}_{i \\in I}$ be subspaces of $V$ and let $W = \\bigcap_{i \\in I} W_i$. - **Non-empty:** Each $W_i$ contains $\\mathbf{0}$, so $\\mathbf{0} \\in W$. - **Closure under addition:** If $u, v \\in W$, then $u, v \\in W_i$ for all $i$. Since each $W_i$ is a subspace, $u + v \\in W_i$ fo... From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.