# MATH1231_assignment (1).pdf - MATH1231ASSIGNMENT 2019 T3...

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MATH1231ASSIGNMENT2019 T3VENKATA APOORVA SMRUTI MANDADI1.Linear MapsThe functionT:R3R2is defined byTx1x2x3=5x1+ 5x33x2+ 3x3for allx1x2x3R3.Show thatTis linear.
Date: October 2019.1
2VENKATA APOORVA SMRUTI MANDADI=-5x1+ 5x33x2+ 3x3+-5y1+ 5y33y2+ 3y3=T(x) +T(y).Therefore, addition is preserved.For Scalar Multiplication:Let,x=x1x2x3be a vector inR3andλbe a scalar value inR. So,λx=λx1λx2λx3.Then,T(λx) =Tλx1λx2λx3=-5λx1+ 5λx33λx2+ 3λx3=λ-5x1+ 5x33x2+ 3x3.Therefore, scalar multiplication is preserved.Since, both the domainR3, and codomainR2are known vector spaces over the same field,and the function satisfies the conditions of preserving addition and scalar multiplication,Tis therefore a Linear transformation.2.Prove a hyperplane is a subspaceShow thatS=xR4: 3x1-x2-2x3-x4= 0is a subspace.
MATH1231ASSIGNMENT2019 T33Check ifScontains the zero vector:Scontains the zero vector,0= (0,0,0,0) since 3(0)-(0)-2(0)-(0) = 0.check if closed under addition:

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Term
Three
Professor
Mak
Tags
x, Find S
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