2. For a linear function f: R" Rm, which of the following statements is/are true? Select all that apply. ☐ Take n = 3, m = 1 and V₁ = [1, 1, 1]¹, v₂ = [1,0, 1], v3 = [2, 1, 2]T. Suppose f(V₁) = 1 and f(v₂) = -3. We must have f (v3) = -1 There exists matrix A in Rmxn so that for any x in R", f(x) = Ax. For zero vectors On in R" and Om in Rm, we have f (On) = 0m. For any x in R", define g(x) = 2x + c, where c is a non-zero vector in R". Then g is a linear function defined on g : R" → Rn.
2. For a linear function f: R" Rm, which of the following statements is/are true? Select all that apply. ☐ Take n = 3, m = 1 and V₁ = [1, 1, 1]¹, v₂ = [1,0, 1], v3 = [2, 1, 2]T. Suppose f(V₁) = 1 and f(v₂) = -3. We must have f (v3) = -1 There exists matrix A in Rmxn so that for any x in R", f(x) = Ax. For zero vectors On in R" and Om in Rm, we have f (On) = 0m. For any x in R", define g(x) = 2x + c, where c is a non-zero vector in R". Then g is a linear function defined on g : R" → Rn.
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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![2. For a linear function f: R" → Rm, which of the following statements is/are true? Select all that apply.
☐ Take n = 3, m = 1 and v₁ = [1,1, 1]¹, v₂ = [1,0, 1], v3 = [2, 1, 2] T. Suppose f(v₁) = 1 and f(v₂) = -3. We must have f(v3) = -1
There exists matrix A in Rmxn so that for any x in R", f(x) = Ax.
For zero vectors On in R" and Om in Rm, we have f (On) = Om.
☐ For any x in R", define g(x) = 2x + c, where c is a non-zero vector in R". Then g is a linear function defined on g : R" → Rn.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6d7ff04-530d-4821-aeda-e8b498d581d8%2F7441bc7c-eb9d-47b0-992a-b638b6e06ac3%2Fa1tcqtk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. For a linear function f: R" → Rm, which of the following statements is/are true? Select all that apply.
☐ Take n = 3, m = 1 and v₁ = [1,1, 1]¹, v₂ = [1,0, 1], v3 = [2, 1, 2] T. Suppose f(v₁) = 1 and f(v₂) = -3. We must have f(v3) = -1
There exists matrix A in Rmxn so that for any x in R", f(x) = Ax.
For zero vectors On in R" and Om in Rm, we have f (On) = Om.
☐ For any x in R", define g(x) = 2x + c, where c is a non-zero vector in R". Then g is a linear function defined on g : R" → Rn.
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