Math 206 Graded Problems/Proofs #20 Due Monday April 22

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Let B = {v₁, v₂, ...,v_n } be a basis for a vector space V and let T:V-->W be a linear transformation. (As usual, W is also to be a vector space.)

Prove: If T(v₁) =T(v₂) = ... =T(v_n) = 0 (the zero vector in W), then T is the zero transformation (from V to W).

A couple of comments.

1. As noted in the text, T:V-->W is the zero transformation from V to W iff T(v)=0 for every vector v in V.
2. You will probably realize that this is a special case of a more general result that I described, in loose terms, in class, but I want you to prove this result without quoting that more general principle.
3. This will be a 5-point proof that should be easily manageable by Monday.

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: April 18, 2001
• Expires: June 30, 2001