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Define f: R→ R by f(x) = mx + b. a. Show that f is a linear transformation when b = 0. b. Find a property of a linear transformation that is violated when b 0. c. Why is f called a linear function?

Define f: R→ R by f(x) = mx + b. a. Show that f is a linear transformation when b = 0. b. Find a property of a linear transformation that is violated when b 0. c. Why is f called a linear function?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
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29
and let aps x into is Ax for se. Justify ction. on defined ansforma- mation. T(C₁V₁+ 2 in the nation. the set of d if c is he range of vector tion of a reflects -lem 2.) b. The line segment from p to q is the set of points of the form (1-1)p+tq for 0 ≤ ≤ 1 (as shown in the figure below). Show that a linear transformation T maps this line segment onto a line segment or onto a single point. (t = 1) q (1-t)p+tq X (t = 0) p 41300 28. Let u and y be vectors in R". It can be shown that the set P of all points in the parallelogram determined by u and v has the form au + bv, for 0 ≤ a ≤ 1,0 ≤ b ≤ 1. Let T: R → Rm be a linear transformation. Explain why the image of a point in P under the transformation T lies in the parallelogram determined by T(u) and T(v). 29. Define f: R → R by f(x) = mx + b. a. Show that f is a linear transformation when b = 0. b. Find a property of a linear transformation that is violated when b 0. Why is f called a linear function? c. 30. An affine transformation T:R" → Rm has the form T(x) = Ax + b, with A an m x n matrix and b in Rm. Show that T is not a linear transformation when b transformations are important in computer graphics.) 0. (Affine 31. Let T: R" → Rm be a linear transformation, and let {V1, V2, V3} be a linearly dependent set in R". Explain why the set {T(v₁), T(v2), T(v3)} is linearly dependent. In Exercises 32-36, column vectors are written as rows, such as x = (x₁, x₂), and T(x) is written as T(x₁, x₂). 32. Show that the transformation T defined by T(x₁, x₂) = (4x12x2, 3|x2) is not linear.
Transcribed Image Text:and let aps x into is Ax for se. Justify ction. on defined ansforma- mation. T(C₁V₁+ 2 in the nation. the set of d if c is he range of vector tion of a reflects -lem 2.) b. The line segment from p to q is the set of points of the form (1-1)p+tq for 0 ≤ ≤ 1 (as shown in the figure below). Show that a linear transformation T maps this line segment onto a line segment or onto a single point. (t = 1) q (1-t)p+tq X (t = 0) p 41300 28. Let u and y be vectors in R". It can be shown that the set P of all points in the parallelogram determined by u and v has the form au + bv, for 0 ≤ a ≤ 1,0 ≤ b ≤ 1. Let T: R → Rm be a linear transformation. Explain why the image of a point in P under the transformation T lies in the parallelogram determined by T(u) and T(v). 29. Define f: R → R by f(x) = mx + b. a. Show that f is a linear transformation when b = 0. b. Find a property of a linear transformation that is violated when b 0. Why is f called a linear function? c. 30. An affine transformation T:R" → Rm has the form T(x) = Ax + b, with A an m x n matrix and b in Rm. Show that T is not a linear transformation when b transformations are important in computer graphics.) 0. (Affine 31. Let T: R" → Rm be a linear transformation, and let {V1, V2, V3} be a linearly dependent set in R". Explain why the set {T(v₁), T(v2), T(v3)} is linearly dependent. In Exercises 32-36, column vectors are written as rows, such as x = (x₁, x₂), and T(x) is written as T(x₁, x₂). 32. Show that the transformation T defined by T(x₁, x₂) = (4x12x2, 3|x2) is not linear.
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