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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).

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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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libo Make two sketches similar to Figure 6 that illustrate prop- noterties (i) and (ii) of a linear transformation. 24. 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 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 ulov 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 > 3 SERA 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? in R" 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. SV (t = 1)q (1-t)p+tq mi GAT SM X (t = 0) p beogami codw
Transcribed Image Text:libo Make two sketches similar to Figure 6 that illustrate prop- noterties (i) and (ii) of a linear transformation. 24. 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 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 ulov 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 > 3 SERA 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? in R" 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. SV (t = 1)q (1-t)p+tq mi GAT SM X (t = 0) p beogami codw
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