Which of the following are linear transformations? (Choose all that apply)OT:RR defined by T(x) == 3.OT:R→R defined by T(x) = x+5.OT: R² →R² defined by T(< x,y >) =< 2y, x − y >.OT: Pn →Pn-1 defined by T(p) = p where P is the vector space of polynomials of degree at most n and p' isthe derivative of p.

(.) Linear transformation: Let and be two vector spaces over a same field , then function is…

Answered: Which of the following are linear…

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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Related questions

Question

Which of the following are linear transformations? (Choose all that apply)
OT:RR defined by T(x) =
= 3.
OT:R→R defined by T(x) = x+5.
OT: R² →R² defined by T(< x,y >) =< 2y, x − y >.
OT: Pn →Pn-1 defined by T(p) = p where P is the vector space of polynomials of degree at most n and p' is
the derivative of p.

Expert Solution

Step 1: We give definition of linear transformation.

(.) Linear transformation:

Let $U$ and $V$ be two vector spaces over a same field $F$ , then function $T space colon space U rightwards arrow space V$ is called a linear transformation if,

$left parenthesis i right parenthesis space space space T open parentheses space u subscript 1 plus u subscript 2 close parentheses space equals space T open parentheses u subscript 1 close parentheses plus T open parentheses u subscript 2 close parentheses space space space space f o r space a l l space u subscript 1 comma u subscript 2 element of U left parenthesis i i right parenthesis space space space space T left parenthesis a u right parenthesis space equals space a T left parenthesis u right parenthesis space space space f o r space a l l space a element of F space comma space u element of U$

Now we find which of the following given are linear transformations.