linear transformation

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We2592

#1 h Apr 25, 2025, 4:09 AM

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Q) let $V$ be one dimensional vector space over field $\mathbb{F}$ then find all linear mapps possible on
$T:V \to V$ ? generalize it for n dimensional ?

Q)let $T:\mathbb{R}^4 \to \mathbb{R}$ be given by $T(x_1,x_2,...,x_n)=x_i$ for a fixed i. then show that $T_i$ is a linear map? generalize it.

Q)let ${\{v_i}\}_{i=1}^{n}$ be a basis of $V$ .Define $T_i:V \to \mathbb{R}$ by $T_i(v)=a_i$ if $v=a_1v_1+...+a_nv_n$ then show that $T_i$ is a linear map.

Q)let $f_i:\mathbb{R}^m \to \mathbb{R}$ be arbitary functions. let $T:\mathbb{R}^m \to \mathbb{R}^n$ be defined by $T(x_1,...,x_m)=(f_1(x),...,f_n(x))$ , when $T$ is linear?

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