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https://texercises.com/exercise/linear-maps-and-span/
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Exercise:
Let T:Vlongrightarrow W be a linear map. Let v_...v_n be a basis of V. Then textImTtextSpTv_...Tv_n.

Solution:
Proof. We first show textSpTv_...Tv_nsubseteq textImT. Let a_...a_nin K. Then _i^n a_iTv_iTleft_i^n a_iv_irightin textImT This shows that textSpTv_...Tv_nsubseteq textImT We now show the opposite direction so that textImTsubseteq textSpTv_...Tv_n. Let win textImT. By definition exists vin V s.t. Tvw. Since v_...v_n is a basis for V exists a_...a_nin K st. v_i^n a_iv_i &Longrightarrow wTvTleft_i^n a_iv_iright _i^n a_iTv_i &Longrightarrow win textSpTv_...Tv_n So textImTsubseteq textSpTv_...Tv_n.
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\(\LaTeX\)-Code
Exercise:
Let T:Vlongrightarrow W be a linear map. Let v_...v_n be a basis of V. Then textImTtextSpTv_...Tv_n.

Solution:
Proof. We first show textSpTv_...Tv_nsubseteq textImT. Let a_...a_nin K. Then _i^n a_iTv_iTleft_i^n a_iv_irightin textImT This shows that textSpTv_...Tv_nsubseteq textImT We now show the opposite direction so that textImTsubseteq textSpTv_...Tv_n. Let win textImT. By definition exists vin V s.t. Tvw. Since v_...v_n is a basis for V exists a_...a_nin K st. v_i^n a_iv_i &Longrightarrow wTvTleft_i^n a_iv_iright _i^n a_iTv_i &Longrightarrow win textSpTv_...Tv_n So textImTsubseteq textSpTv_...Tv_n.
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eth, hs22, image, linear map, lineare algebra, proof, span, vector space
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(3, default)
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ENG (English)
Type
Proof
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