Question

Let U and V be vector spaces over a field F,a function T:U→V
such that
T(u1+u2)=T(u1)+T(u2), for u1,u2∈U and T(αu)=αT(u) for α∈F and u∈U
is called a ………….

(A) Vector space
(B) Transformation
(C) Linear transformation
(D) Nullity

Answer 1

Question #22433 Let $U$ and $V$ be vector spaces over a field $F$, a function $T\colon U\to F$, such that $T(\alpha_{1}u_{1}+\alpha_{2}u_{2})=\alpha_{1}T(u_{1})+\alpha_{2}T(u_{2})$, for $\alpha_{1},\alpha_{2}\in F$ and $u_{1},u_{2}\in U$

is a called a … if (A) Vector space

(B) Transformation

(C) Linear transformation

(D) Nullity

Please explain

Solution. By definition the property of function $T$: $T(\alpha_{1}u_{1}+\alpha_{2}u_{2})=\alpha_{1}T(u_{1})+\alpha_{2}T(u_{2})$, for $\alpha_{1},\alpha_{2}\in F$ and $u_{1},u_{2}\in U$ is linearity property. Vector space is not a function, as well as nullity, which is dimension of kernel of $T$. Transformation is synonym to “function”, however this does not describe the property in question Hence

Answer C.

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