https://gdpark.blog/series/linear-algebra-i-studied/Recent content in Linear Algebra I Studied on gdpark.blogHugoenTue, 02 Feb 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-01-prologue-heralding-the-beginning/Mon, 18 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-01-prologue-heralding-the-beginning/Stumbled onto a hidden-gem linear algebra course on educast.pro while cramming for quantum mechanics, and the pure love-of-learning vibe had me completely hooked.https://gdpark.blog/posts/linear-algebra-01-vector-spaces/Mon, 18 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-01-vector-spaces/Turns out &ldquo;linear&rdquo; literally just means line-shaped — lol — and here&rsquo;s why that embarrassingly simple idea is the backbone of basically all of math and STEM.https://gdpark.blog/posts/linear-algebra-02-linear-dependence-and-linear-independence/Tue, 19 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-02-linear-dependence-and-linear-independence/Breaking down linear combinations, span, and what it actually means for vectors to be linearly independent — all in plain terms.https://gdpark.blog/posts/linear-algebra-03-basis/Tue, 19 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-03-basis/So a basis is literally the intersection of spanning sets and linearly independent sets — and there are TONS of them, plus order actually matters!!https://gdpark.blog/posts/linear-algebra-04-matrices-and-matrix-multiplication/Wed, 20 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-04-matrices-and-matrix-multiplication/We dive into linear algebra by unpacking functions as set mappings, then zoom in on linear mappings and how they all secretly boil down to matrices.https://gdpark.blog/posts/linear-algebra-05-surjective-injective-and-bijective-functions/Wed, 20 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-05-surjective-injective-and-bijective-functions/We nail down surjective, injective, and bijective functions — and prove why same-dimension linear maps only need one condition to guarantee the other.https://gdpark.blog/posts/linear-algebra-06-kernel-and-image/Thu, 21 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-06-kernel-and-image/We nail down the kernel and image of a linear map, see what injective and surjective really mean, and lock in the theorem that links them all!!!!https://gdpark.blog/posts/linear-algebra-07-isomorphism-and-the-dimension-theorem/Thu, 21 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-07-isomorphism-and-the-dimension-theorem/Kernel and image lead us to isomorphism — two vector spaces with the same dimension and a bijective map between them are literally just the same space in disguise!!!!https://gdpark.blog/posts/linear-algebra-08-change-of-basis/Fri, 22 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-08-change-of-basis/Same linear map, totally different matrix depending on your basis — let&rsquo;s dig into why that happens and how A and A&rsquo; are actually related.https://gdpark.blog/posts/linear-algebra-09-systems-of-linear-equations-and-gaussian-elimination/Fri, 22 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-09-systems-of-linear-equations-and-gaussian-elimination/A casual walkthrough of systems of linear equations — from middle-school basics all the way up to matrix form, homogeneous vs. non-homogeneous, and Gaussian elimination.https://gdpark.blog/posts/linear-algebra-10-elementary-row-operation-matrices-erom/Sat, 23 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-10-elementary-row-operation-matrices-erom/Back at it with the general solution of AX = B — recapping RREF, breaking down dependent vs. independent variables, and chasing down X_0 and ker A.https://gdpark.blog/posts/linear-algebra-11-the-rank-theorem/Sat, 23 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-11-the-rank-theorem/We poke at whether RREF is unique, then laser in on invertible matrices to show their RREF has to be the identity — and that&rsquo;s the Rank Theorem taking shape!https://gdpark.blog/posts/linear-algebra-12-multilinear-forms-alternating-forms-and-the-epsilon-symbol/Sun, 24 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-12-multilinear-forms-alternating-forms-and-the-epsilon-symbol/We finally tackle <em>how</em> to tell if a matrix is invertible by wading through multilinear forms, alternating forms, and the epsilon symbol — all the groundwork for determinants!https://gdpark.blog/posts/linear-algebra-13-similar-matrices/Sun, 24 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-13-similar-matrices/Before we can actually crunch a determinant, we load up on similar matrices — plus why det A = 0 is the snap-decision test for invertibility.https://gdpark.blog/posts/linear-algebra-14-determinants/Mon, 25 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-14-determinants/We finally crack the determinant open — from 2x2 all the way to 4x4 — by tracing it straight back to where it really came from: the Alternating Form, lol.https://gdpark.blog/posts/linear-algebra-15-the-adjoint-matrix/Tue, 26 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-15-the-adjoint-matrix/We dig into triangular and Vandermonde matrices, crack their determinants, then build up to the adjoint matrix — which is about to become super important for what&rsquo;s coming next.https://gdpark.blog/posts/linear-algebra-16-block-matrices/Wed, 27 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-16-block-matrices/A casual dive into diagonalization — what it means to turn a matrix diagonal, why that&rsquo;s such a big deal in physics, and how blocking gets the whole thing started.https://gdpark.blog/posts/linear-algebra-17-introduction-to-diagonalization-eigenvalues-and-eigenvectors/Thu, 28 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-17-introduction-to-diagonalization-eigenvalues-and-eigenvectors/Finally cracking how to diagonalize a matrix without hunting for the right basis — turns out eigenvalues and eigenvectors are the secret the whole time!https://gdpark.blog/posts/linear-algebra-18-characteristic-polynomials-and-the-cayley-hamilton-theorem/Fri, 29 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-18-characteristic-polynomials-and-the-cayley-hamilton-theorem/We chase down which matrices can actually be diagonalized — starting with characteristic polynomials, hitting complex eigenvalues, and landing on the Cayley–Hamilton theorem.https://gdpark.blog/posts/linear-algebra-19-number-theory-and-polynomial-background-for-the-decompositio/Sat, 30 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-19-number-theory-and-polynomial-background-for-the-decompositio/A casual walk through naturals, integers, divisibility, and the division algorithm, building up to why any subset of Z closed under +, −, and × must look like nZ.https://gdpark.blog/posts/linear-algebra-20-the-first-decomposition-theorem/Sun, 31 Jan 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-20-the-first-decomposition-theorem/We revisit diagonalization up close — eigenvalues, kernels, and how a 2D space splits into two 1D invariant subspaces — as a warm-up for the full Decomposition Theorem.https://gdpark.blog/posts/linear-algebra-21-minimal-polynomials-companion-matrices-and-more/Mon, 01 Feb 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-21-minimal-polynomials-companion-matrices-and-more/Diving into minimal polynomials and their sneaky relationship with characteristic polynomials — turns out it&rsquo;s the same divisor-hunting logic we used back in number theory!!!!https://gdpark.blog/posts/linear-algebra-22-is-further-block-decomposition-always-possible/Tue, 02 Feb 2016 00:00:00 +0000https://gdpark.blog/posts/linear-algebra-22-is-further-block-decomposition-always-possible/Poking at a special case where the minimal and characteristic polynomials match as a power of an irreducible polynomial, and what that tells us about block decomposition.