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=x∈R4: 3x1-x2-2x3-x4= 0is a subspace.
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