let into for stify ined ma- on. Vit the on. et of Ic is ange ector n of a flects ises 21 and 22 in Section 1.5.) b. The line segment from p to q is the set of points of the form (1-t)p+tq for 0 ≤ t ≤ 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 28. Let u and v 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" → R™ 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. c. Why is f called a linear function? 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) = lu Dig not linear
let into for stify ined ma- on. Vit the on. et of Ic is ange ector n of a flects ises 21 and 22 in Section 1.5.) b. The line segment from p to q is the set of points of the form (1-t)p+tq for 0 ≤ t ≤ 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 28. Let u and v 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" → R™ 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. c. Why is f called a linear function? 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) = lu Dig not linear
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
29
Transcribed Image Text:let
into
for
stify
ined
ma-
on.
Vit
the
on.
et of
Ic is
ange
ector
n of a
flects
ises 21 and 22 in Section 1.5.)
b. The line segment from p to q is the set of points of the
form (1-t)p+tq for 0 ≤ t ≤ 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
28. Let u and v 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" → R™
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.
c. Why is f called a linear function?
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
th T is not a linear transformation when b# 0. (Affine
transformations are important in computer graphics.)
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) =
alu Din not linear
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