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5 Is Mmxn Mnxm?

5 Is Mmxn Mnxm?

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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Please do Exericse 2.15 please show a proof with your answer. Please show step by step and explain

In Chapter Two, with the definition of vector spaces, we seemed to have opened up our studies to many examples of new structures besides the familiar R¹'s. We now know that isn't the case. Any finite-dimensional vector space is actually "the same" as a real space. Exercises ✓2.10 Decide if the spaces are isomorphic. (a) R², R4 (b) P5, R5 (c) M2x3, R6 (e) M2xk, M₁x2 (d) P5, M2x3 2.11 Which of these spaces are isomorphic to each other? (a) R³ (b) M₂x2 (c) P3 (d) R4 (e) P₂ ✓2.12 Consider the isomorphism RepB (-): P₁ → R² where B image of each of these elements of the domain. (a) 3-2x; (b) 2+2x; (c) x (1,1+x). Find the 2.13 For which n is the space isomorphic to R"? (a) P₁ (b) P₁ (c) M2x3 (d) the plane 2x -y +z = 0 subset of R³ (e) the vector space of linear combinations of three letters {ax+by+cz | a, b, c = R} ✓2.14 Show that if m n then R™ #R". ✓2.15 Is Mmxn Mnxm? ✓2.16 Are any two planes through the origin in R³ isomorphic?
Transcribed Image Text:In Chapter Two, with the definition of vector spaces, we seemed to have opened up our studies to many examples of new structures besides the familiar R¹'s. We now know that isn't the case. Any finite-dimensional vector space is actually "the same" as a real space. Exercises ✓2.10 Decide if the spaces are isomorphic. (a) R², R4 (b) P5, R5 (c) M2x3, R6 (e) M2xk, M₁x2 (d) P5, M2x3 2.11 Which of these spaces are isomorphic to each other? (a) R³ (b) M₂x2 (c) P3 (d) R4 (e) P₂ ✓2.12 Consider the isomorphism RepB (-): P₁ → R² where B image of each of these elements of the domain. (a) 3-2x; (b) 2+2x; (c) x (1,1+x). Find the 2.13 For which n is the space isomorphic to R"? (a) P₁ (b) P₁ (c) M2x3 (d) the plane 2x -y +z = 0 subset of R³ (e) the vector space of linear combinations of three letters {ax+by+cz | a, b, c = R} ✓2.14 Show that if m n then R™ #R". ✓2.15 Is Mmxn Mnxm? ✓2.16 Are any two planes through the origin in R³ isomorphic?
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