Let lam E N. Let 171,-',1719 be some basis for R''. a) Suppose that L : Rk > R' and T : W' > Rm are two linear transformations . Suppose that
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Let k,m ∈ N. Let ⃗v1,...,⃗vk be some basis for Rk.

  1. a) Suppose that L : Rk → Rm and T : Rk → Rm are two linear transformations . Suppose that L(w⃗) ̸= T(w⃗) for some vector w⃗ ∈ Rk. Show that there must exist some i ∈ {1,...,k} such that the functions L and T disagree on the ith basis vector; that is, L(⃗vi) ̸= T(⃗vi).
  2. b) Suppose that every element in Rm is in the image of L. Can you conclude that {L(⃗v1), . . . , L(⃗vk)}
  3. is a basis for Rm? Prove it or find a counterexample.

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Let lam E N. Let 171,-",1719 be some basis for R'". a) Suppose that L : Rk —> R" and T : W" —> Rm are two linear transformations . Suppose that L(7.D') aé TOE) for some vector 13 E R". Show that there must exist some 21 E {1, . . . ,k} such that the functions L and T disagree on the ith basis vector; that is, LUZ) 7E T(17,;). b) Suppose that every element in Rm is in the image of L. Can you conclude that {L(1_)'1),. . . , L(fik)} is a basis for Rm? Prove it or find a counterexample.

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Subject: Linear Algebra, Math
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