Solution Found!
Let u and v be linearly independent vectors in ?3, and let
Chapter 1, Problem 26E(choose chapter or problem)
PROBLEM 26E
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 + t v (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?
Questions & Answers
QUESTION:
PROBLEM 26E
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 + t v (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?
ANSWER:
Solution
Step 1:
Given : 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 + t v (with s, t in ℝ)
We have to 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.
And What must be true about T (u) and T (v) in order for the image of the plane P to be a plane?