Suppose (vi) Show that T is the zero transformation. 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 U for = 1,..., P. That is, show that if THOBI
Suppose (vi) Show that T is the zero transformation. 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 U for = 1,..., P. That is, show that if THOBI
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
Need answer to 25 only please
Transcribed Image Text:Make two sketches similar to Figure 6 that illustrate prop-
erties (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
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 7 : 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 7(v) in order for the image of
3
the plane P to
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