Make two sketches similar to Figure 6 that illustrate prop- erties (i) and (ii) of a linear transformation. Suppose vectors V₁,..., Vp span R", and let T : R" → R" be a linear transformation. Suppose T (vi) = 0 for i = 1, ..., p. Show that T is the zero transformation. That is, show that if x is any vector in R", then 7(x) = 0. = Given v 0 and p in R", the line through p in the direction of v has the parametric equation x p+ tv. Show that a linear transformation T: R" →R" maps this line onto another line or onto a single point (a degenerate line). Let u and v be linearly independent vectors in R³, and let P be the plane through u, v, and 0. The parametric equation of P is x = su + tv (with s,t in R). Show that a linear transformation T: R³ → R³ maps P onto a plane through 0, or onto a line through 0, or onto just the origin in R³. What must be true about T(u) and T(v) in order for the image of the plane P to be a plane? To $1 a. Show that the line through vectors p and q in R" may be written in the parametric form x = (1-t)p+tq. (Refer to the figure with Exercises 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 7 maps this ling coar
Make two sketches similar to Figure 6 that illustrate prop- erties (i) and (ii) of a linear transformation. Suppose vectors V₁,..., Vp span R", and let T : R" → R" be a linear transformation. Suppose T (vi) = 0 for i = 1, ..., p. Show that T is the zero transformation. That is, show that if x is any vector in R", then 7(x) = 0. = Given v 0 and p in R", the line through p in the direction of v has the parametric equation x p+ tv. Show that a linear transformation T: R" →R" maps this line onto another line or onto a single point (a degenerate line). Let u and v be linearly independent vectors in R³, and let P be the plane through u, v, and 0. The parametric equation of P is x = su + tv (with s,t in R). Show that a linear transformation T: R³ → R³ maps P onto a plane through 0, or onto a line through 0, or onto just the origin in R³. What must be true about T(u) and T(v) in order for the image of the plane P to be a plane? To $1 a. Show that the line through vectors p and q in R" may be written in the parametric form x = (1-t)p+tq. (Refer to the figure with Exercises 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 7 maps this ling coar
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
25
Transcribed Image Text:bo Make two sketches similar to Figure 6 that illustrate prop-
Botterties (i) and (ii) of a linear transformation.
→R" be
24. Suppose vectors V₁, ..., V, span R", and let T : R"
a linear transformation. Suppose T(vi) = 0 for i = 1, ..., p.
Show that T is the zero transformation. That is, show that if
x is any vector in R", then 7(x) = 0.
25. Given v 0 and p in R", the line through p in the direction of
v has the parametric equation x = p + tv. Show that a linear
transformation T: R" → R" maps this line onto another line
or onto a single point (a degenerate line).
26. Let u and v be linearly independent vectors in R³, and let P
be the plane through u, v, and 0. The parametric equation
of P is x = su + tv (with s,t in R). Show that a linear
transformation T: R³ →> R3 maps P onto a plane through
0, or onto a line through 0, or onto just the origin in R³. What
must be true about T(u) and T(v) in order for the image of
the plane P to be a plane?
To $1
27. a. Show that the line through vectors p and q in R" may be
written in the parametric form x = (1-t)p+tq. (Refer
to the figure with Exercises 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
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