Algebra
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This document contains solutions to exercises on linear transformations. It begins with objectives to analyze and calculate linear transformations and develop skills from class. It then shows work solving 7 exercises involving determining if functions define linear transformations, calculating outputs of transformations given inputs, and determining an output given transformation definitions on other inputs. References include YouTube videos on linear transformations.
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Taller 1 parcial 3
This document discusses three exercises involving linear transformations. The exercises ask the reader to determine if given functions define linear transformations and to determine the output of linear transformations given their behavior on sample inputs. The document provides the definitions, inputs, and step-by-step workings to solve each exercise. It concludes that exercises 1 and 3 define linear transformations while exercise 2 does not and determines the output of two other linear transformations.
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The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
The third function f(x,y,z) = (x +
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2) For the function y = 2^3x, the logarithmic derivative is y'/y = 3/2x.
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3. Factoring polynomials into irreducible factors.
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This document discusses three exercises involving linear transformations. The exercises ask the reader to determine if given functions define linear transformations and to determine the output of linear transformations given their behavior on sample inputs. The document provides the definitions, inputs, and step-by-step workings to solve each exercise. It concludes that exercises 1 and 3 define linear transformations while exercise 2 does not and determines the output of two other linear transformations.
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The document analyzes linear transformations in R2 and R3. It determines whether three given functions define linear transformations based on whether they satisfy the property T(αu + βv) = αT(u) + βT(v).
The first function f(x,y) = (3x - y, x + y) is determined to define a linear transformation in R2 since it satisfies the property.
The second function f(x,y,z) = (x,y,z2) is determined not to define a linear transformation in R3 since z2 is not a linear term.
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The document discusses singular value decomposition (SVD), which is a way to decompose a matrix A into three matrices: A = UΣV^T. U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A. SVD can be used to perform dimensionality reduction by approximating A using only the top k singular values/vectors in Σ, U, and V^T. This reduces the number of parameters needed to represent A while retaining most of its information.
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These will be helpful for class 9-10 and 11-12 students, especially in Bangladesh. Here, I documented all the important formulas required for solving infinite series problems in mathematics.
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The document discusses the derivation of the logarithmic derivative of several functions:
1) For the function y = xln(x), the logarithmic derivative is y'/y = 1/x.
2) For the function y = 2^3x, the logarithmic derivative is y'/y = 3/2x.
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This document provides information on factoring polynomials with a common monomial factor. It defines a common monomial factor as a number, variable, or combination that appears in each term. It outlines the steps to factor polynomials with this common factor: find the greatest common factor (GCF), divide the polynomial by the GCF, and express the factorization. Examples are provided to demonstrate this process. Students are then assigned practice problems to factor polynomials using this method and given a deadline to submit their work.
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The document discusses linear transformations. It provides examples of determining if functions define linear transformations by checking if they satisfy the property that T(αu + βv) = αT(u) + βT(v). It then gives an example of using a system of equations to determine the output of a linear transformation T for a given input, when the outputs of T for two other example inputs are given.
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This document provides examples of using partial fractions to decompose rational functions into simpler forms that can be integrated term-by-term. It reviews the steps to factor the denominator completely, then make fractions for each linear and repeated linear factor in the form A/(x-c) and for each quadratic factor in the form (Ax+B)/(x^2-c). Examples are worked through, showing the decomposition, substitution to solve for coefficients, and integration of each term. Shortcuts are noted when the denominator factors into linear terms only.
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ملزمة الرياضيات للصف السادس العلمي الاحيائي - التطبيقي
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Null Bangalore | Pentesters Approach to AWS IAM
#Abstract:
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#Prerequisites:
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Algebra
- 1. DEPARTAMENTODE CIENCIAS EXACTAS ALGEBRALINEAL PARCIALII TALLERNO 3 TEMA: TRANSFORMACIONES LINEALES INTEGRANTES: - GUERRERO CÓRDOVA STEFANY PAOLA - SUNTA BARZALLO JOSSELYN GABRIELA - UNDA ROBLRES GRACE IVETTE GRUPON 4 NRC:4264 FECHADE ENTREGA:4/3/2021
- 2. ÍNDICE OBJETIVO…………………………………………………………3 SOLUCIÓN EJERCICIO 1)………………………………………4 • SOLUCIÓN EJERCICIO 2) ……………………………………………..5 • SOLUCIÓN EJERCICIO 3)………………………………………………6 • SOLUCIÓN EJERCICIO 6)………………………………………………7 • SOLUCION EJERCICIO 6)………………………………………………8 • SOLUCIÓN EJERCICIO 7)………………………………………………9 • BIBLIOGRAFÍA …………………………………………………………….10
- 3. OBJETIVOS: • Analizar y calcular las transformaciones lineales. • Desarrollar las destrezas obtenidas en hora clase.
- 4. DETERMINAR CUÁL DE LAS SIGUIENTES FUNCIONES, DEFINE UNA TRANSFORMACIÓN LINEAL 𝟏. 𝒇 𝒙, 𝒚 = 𝟑(𝒙 − 𝒚, 𝒙 + 𝒚) 𝑢 = 𝑥1 𝑦2 𝑣 = 𝑥2 𝑦2 𝑇 𝛼𝑢 + 𝛽𝑣 = 𝑇 𝛼 𝑥1 𝑦1 + 𝛽 𝑥2 𝑦2 = 𝑇 𝛼𝑥1 + 𝛽𝑦1 𝛼𝑥2 + 𝛽𝑦2 = 3 𝛼𝑥1 + 𝛽𝑥2 − (𝛼𝑦1 + 𝛽𝑦2) 𝛼𝑥1 + 𝛽𝑥2 + (𝛼𝑦1 + 𝛽𝑦2) = 3 𝛼 𝑥1 − 𝑦1 + 𝛽(𝑥2 − 𝑦2) 𝛼 𝑥1 + 𝑦1 + 𝛽(𝑥2 + 𝑦2) = 3𝛼 𝑥1 − 𝑦1 𝑥1 + 𝑦1 + 3𝛽 𝑥2 − 𝑦2 𝑥2 + 𝑦2 = 3 𝛼𝑢 + 𝛽𝑣 = 3 𝛼𝑇 𝑢 + 𝛽𝑇 𝑣 → 𝑺𝒊𝒆𝒔 𝒖𝒏𝒂 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎𝒂𝒄𝒊ó𝒏 𝑳𝒊𝒏𝒆𝒂𝒍. 𝑆𝑖 𝑒𝑠 𝑢𝑛𝑎 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝐿𝑖𝑛𝑒𝑎𝑙.
- 5. 𝟐. 𝒇 𝒙, 𝒚, 𝒛 = (𝒙, 𝒚, 𝒛𝟐) 𝑢 = 𝑥1 𝑦1 𝑧1 𝑣 = 𝑥2 𝑦2 𝑧2 𝑇 𝛼𝑢 + 𝛽𝑣 = 𝑇 𝛼 𝑥1 𝑦1 𝑧1 + 𝛽 𝑥2 𝑦2 𝑧2 = 𝑇 𝛼𝑥1 + 𝛽𝑥2 𝛼𝑦1 + 𝛽𝑦2 𝛼𝑧1 + 𝛽𝑧2 = 𝛼𝑥1 + 𝛽𝑥2 𝛼𝑦1 + 𝛽𝑦2 (𝛼𝑧1 + 𝛽𝑧2)2 = 𝛼𝑥1 + 𝛽𝑥2 𝛼𝑦1 + 𝛽𝑦2 (𝛼𝑧1)2+2𝑧1𝑧2𝛼𝛽 + (𝛽 𝑧2 )2 ≠ 𝑥 𝑦 𝑧2 𝑁𝑜 𝑒𝑠 𝑢𝑛𝑎 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝐿𝑖𝑛𝑒𝑎𝑙.
- 6. 𝟑. 𝒇((𝒙, 𝒚, 𝒛)) = (𝒙 + 𝟐𝒚 – 𝟑𝒛, 𝟑𝒙 − 𝒚 + 𝟓𝒛, 𝒙 – 𝒚 – 𝒛) U= 𝑥1 𝑦1 𝑧1 V= 𝑥2 𝑦2 𝑧2 T(𝛼𝑢 + 𝛽𝑣)=T 𝛼 𝑥1 𝑦1 𝑧1 + 𝛽 𝑥2 𝑦2 𝑧2 T 𝛼𝑥1 + 𝛽𝑥2 𝛼𝑦1 + 𝛽𝑦2 𝛼𝑧1 + 𝛽𝑧2 = 𝛼𝑥1 + 𝛽𝑥2 + 2 𝛼𝑦1 + 𝛽𝑦2 − 3(𝛼𝑧1 + 𝛽𝑧2) 3 𝛼𝑥1 + 𝛽𝑥2 − 𝛼𝑦1 + 𝛽𝑦2 + 5(𝛼𝑧1 + 𝛽𝑧2) 𝛼𝑥1 + 𝛽𝑥2 − 𝛼𝑦1 + 𝛽𝑦2 − (𝛼𝑧1 + 𝛽𝑧2) 𝛼 𝑥1 + 2𝑦1 − 3𝑧1 3𝑧1 − 𝑧1 + 5𝑧1 𝑥1 − 𝑦1 − 𝑧1 + 𝛽 𝑥2 + 2𝑦2 − 3𝑧2 3𝑥2 − 𝑦2 + 5𝑧2 𝑥2 − 𝑦2 − 𝑧2 T(𝛼𝑢 + 𝛽𝑣) =𝛼𝑇 𝑢 + 𝛽𝑇 𝑣 → 𝑺𝒊 𝒆𝒔 𝒖𝒏𝒂 𝒕𝒓𝒂𝒏𝒔𝒇𝒐𝒓𝒎𝒂𝒄𝒊ó𝒏 𝑳𝒊𝒏𝒆𝒂𝒍.
- 7. 6.- Sea f una transformación lineal de 𝑹𝟑 en 𝑹𝟑 , suponga que 𝒇((𝟏, 𝟎, 𝟏)) = (𝟏, −𝟏, 𝟑) 𝒚 𝒇((𝟐, 𝟏, 𝟎)) = (𝟎, 𝟐, 𝟏); Determine 𝒇((−𝟏, −𝟐, 𝟑)). 𝑥 𝑦 𝑧 = 𝛼 1 0 1 + β 2 1 0 + δ −1 −2 3 1 0 1 2 1 0 −1 −2 3 𝑥 𝑦 𝑧 F3-F1 F3 1 0 0 2 1 −2 −1 −2 4 𝑥 𝑦 𝑧 − 𝑥 𝐹3 + 2𝐹2 → 𝐹3 1 0 0 2 1 0 −1 −2 0 𝑥 𝑦 𝑧 − 𝑥 + 2𝑦 𝑇 𝑥 𝑦 𝑧 = 𝑇 1 0 1 + β 2 1 0 + δ −1 −2 3 𝑇 𝑥 𝑦 𝑧 = 𝑇 𝛼 1 0 1 + Tβ 2 1 0 + Tδ −1 −2 3 δ = 0 δ = 0 β - 2 δ = y β – 0 = y β = y 𝛼 + δ − δ = x 𝛼 + 2𝑦 − 0 = x 𝛼 = 𝑥 − 2𝑦
- 8. 𝑇 𝑥 𝑦 𝑧 = 𝑥 − 2𝑦 1 0 1 + y 2 1 0 + 0 −1 −2 3 𝑇 𝑥 𝑦 𝑧 = 𝑥 − 2𝑦 1 −1 3 + y 0 2 1 + 0 𝑁1 𝑁2 𝑁3 𝑇 𝑥 𝑦 𝑧 = 𝑥 − 2𝑦 1 −1 3 + y 0 2 1 𝑇 𝑥 𝑦 𝑧 = 𝑥 − 2𝑦 + 0 −𝑥 + 2𝑦 + 2𝑦 3𝑥 − 6𝑦 + 𝑦 = 𝑥 − 2𝑦 −𝑥 + 4𝑦 3𝑥 − 5𝑦 𝑇 −1 −2 3 = 𝟑 −𝟕 𝟕
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