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https://texercises.com/exercise/dimension-span-linear-independence/
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Exercise:
Suppose ndimV infty. Let v_...v_k be a list of vectors in V. Then: abcliste abc If k n the list v_...v_k can NOT span V. abc If k n the list v_...v_k can NOT be linearly indepent. abcliste

Solution:
Proof. abcliste abc Ase that k n. Suppose by contradiction that Spv_...v_kV. By the mary above we can delete some of the vectors from the list v_...v_k and get a basis v_i_...v_i_k' of V Longrightarrow dimVk'leq k n contradiction lightning. abc Ase k n. Suppose by contradiction that v_...v_k are linearly indepent Longrightarrow we can ext this list to a basis of V Longrightarrow dimV geq k n contradiction lightning. abcliste
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Exercise:
Suppose ndimV infty. Let v_...v_k be a list of vectors in V. Then: abcliste abc If k n the list v_...v_k can NOT span V. abc If k n the list v_...v_k can NOT be linearly indepent. abcliste

Solution:
Proof. abcliste abc Ase that k n. Suppose by contradiction that Spv_...v_kV. By the mary above we can delete some of the vectors from the list v_...v_k and get a basis v_i_...v_i_k' of V Longrightarrow dimVk'leq k n contradiction lightning. abc Ase k n. Suppose by contradiction that v_...v_k are linearly indepent Longrightarrow we can ext this list to a basis of V Longrightarrow dimV geq k n contradiction lightning. abcliste
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dimension, eth, hs22, linear independence, lineare algebra, proof, span
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(3, default)
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ENG (English)
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Proof
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