Cocalc Chapter 7 Linear Independence Ipynb

If all ccc's are zero, a set of vectors {v1,v2,...,vp}\{v_1, v_2,...,v_p\}{v1​,v2​,...,vp​} is said to be linearly independent, if the equation If any of ci≠0c_i\neq 0ci​=0, the set of vectors is linearly dependent. Determine if v1,v2,v3{v}_1, {v}_2, {v}_3v1​,v2​,v3​ are linearly independent. v1=[123]T,v2=[456]T, and v3=[210]T {v}_{1}=\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right]^T, {v}_{2}=\left[\begin{array}{l} 4 \\ 5 \\ 6 \end{array}\right]^T, \text { and } {v}_{3}=\left[\begin{array}{l} 2 \\ 1 \\ 0 \end{array}\right]^T v1​=​123​​T,v2​=​456​​T, and v3​=​210​​T The common way of testing linear combination is to construct augmented matrix and calculate the reduced form, for example ([10−2001100000], (0, 1))\displaystyle \left( \left[\begin{matrix}1 & 0 & -2 & 0\\0 & 1 & 1 & 0\\0 & 0 & 0 & 0\end{matrix}\right], \ \left( 0, \ 1\right)\right)​​100​010​−210​000​​, (0, 1)​

There was an error while loading. Please reload this page. By the end of this comprehensive tutorial, you will: Master linear programming fundamentals and mathematical formulation Understand the geometric interpretation of LP problems and feasible regions Apply duality theory and perform sensitivity analysis

Solve real-world optimization problems in production, transportation, and finance Let {v1,v2,…,vn}\{ v_1, v_2, \ldots, v_n \}{v1​,v2​,…,vn​} be a set of linearly independent vectors in a vector space. Suppose that Then α1=β1,α2=β2,…,αn=βn.\alpha_1 = \beta_1, \alpha_2 = \beta_2, \ldots, \alpha_n = \beta_n\text{.}α1​=β1​,α2​=β2​,…,αn​=βn​. Since v1,…,vnv_1, \ldots, v_nv1​,…,vn​ are linearly independent, αi−βi=0\alpha_i - \beta_i = 0αi​−βi​=0 for i=1,…,n.i = 1, \ldots, n\text{.}i=1,…,n. A set {v1,v2,…,vn}\{ v_1, v_2, \dots, v_n \}{v1​,v2​,…,vn​} of vectors in a vector space VVV is linearly dependent if and only if one of the viv_ivi​'s is a linear combination of the rest.

Suppose that {v1,v2,…,vn}\{ v_1, v_2, \dots, v_n \}{v1​,v2​,…,vn​} is a set of linearly dependent vectors. Then there exist scalars α1,…,αn\alpha_1, \ldots, \alpha_nα1​,…,αn​ such that There was an error while loading. Please reload this page. This work by Jephian Lin is licensed under a Creative Commons Attribution 4.0 International License. \newcommand{\trans}{^\top} \newcommand{\adj}{^{\rm adj}} \newcommand{\cof}{^{\rm cof}} \newcommand{\inp}[2]{\left\langle#1,#2\right\rangle} \newcommand{\dunion}{\mathbin{\dot\cup}} \newcommand{\bzero}{\mathbf{0}} \newcommand{\bone}{\mathbf{1}} \newcommand{\ba}{\mathbf{a}} \newcommand{\bb}{\mathbf{b}} \newcommand{\bc}{\mathbf{c}} \newcommand{\bd}{\mathbf{d}} \newcommand{\be}{\mathbf{e}} \newcommand{\bh}{\mathbf{h}} \newcommand{\bp}{\mathbf{p}} \newcommand{\bq}{\mathbf{q}} \newcommand{\br}{\mathbf{r}} \newcommand{\bx}{\mathbf{x}} \newcommand{\by}{\mathbf{y}} \newcommand{\bz}{\mathbf{z}} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} \newcommand{\bw}{\mathbf{w}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\nul}{\operatorname{null}} \newcommand{\rank}{\operatorname{rank}} %\newcommand{\ker}{\operatorname{ker}} \newcommand{\range}{\operatorname{range}} \newcommand{\Col}{\operatorname{Col}} \newcommand{\Row}{\operatorname{Row}} \newcommand{\spec}{\operatorname{spec}} \newcommand{\vspan}{\operatorname{span}} \newcommand{\Vol}{\operatorname{Vol}}...

Let S={u1,…,uk}S = \{\bu_1,\ldots,\bu_k\}S={u1​,…,uk​} be a set of finite many vectors. We say SSS is linearly independent if the only coefficients c1,…,ck∈Rc_1,\ldots, c_k\in\mathbb{R}c1​,…,ck​∈R satisfying c1u1+⋯+ckuk=0c_1\bu_1 + \cdots + c_k\bu_k = \bzeroc1​u1​+⋯+ck​uk​=0 is c1=⋯=ck=0c_1 = \cdots = c_k = 0c1​=⋯=ck​=0. (A infinite set of vectors is linearly independent if any finite subsut of it is linearly independent.) For any vector u∈S\bu\in Su∈S, u∉span⁡(S∖{u})\bu\notin\vspan(S\setminus\{\bu\})u∈/span(S∖{u}). For any vector u∈S\bu\in Su∈S, span⁡(S∖{u})⊊span⁡(S)\vspan(S\setminus\{\bu\})\subsetneq\vspan(S)span(S∖{u})⊊span(S). There was an error while loading.

Please reload this page. Master Stokes' theorem connecting surface integrals to line integrals through computational practice with curl calculations and circulation analysis. This comprehensive Jupyter notebook demonstrates oriented surfaces with boundary curves, curl interpretation, and applications to electromagnetic theory and fluid dynamics. CoCalc's SageMath environment offers 3D visualization of surfaces and boundaries, symbolic curl computation, and theorem verification tools, enabling students to understand the deep relationship between circulation around boundaries and curl through surfaces. This notebook contains Chapter 7 from the main Advanced Calculus with SageMath notebook. For the complete course, please refer to the main notebook: Advanced Calculus with SageMath.ipynb

This relates the circulation of a vector field around a closed curve to the flux of its curl through any surface bounded by that curve. STOKES’ THEOREM\displaystyle \textbf{STOKES' THEOREM}STOKES’ THEOREM Let $\\{ v_1, v_2, \\ldots, v_n \\}$ be a set of linearly independent vectors in a vector space. Suppose that Then $\\alpha_1 = \\beta_1, \\alpha_2 = \\beta_2, \\ldots, \\alpha_n = \\beta_n\\text{.}$ Since $v_1, \\ldots, v_n$ are linearly independent, $\\alpha_i - \\beta_i = 0$ for $i = 1, \\ldots, n\\text{.}$

A set $\\{ v_1, v_2, \\dots, v_n \\}$ of vectors in a vector space $V$ is linearly dependent if and only if one of the $v_i$'s is a linear combination of the rest. Suppose that $\\{ v_1, v_2, \\dots, v_n \\}$ is a set of linearly dependent vectors. Then there exist scalars $\\alpha_1, \\ldots, \\alpha_n$ such that There was an error while loading. Please reload this page.

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