Let V and W be vector spaces, and let L: V → W be a linear transformation. Let B = {U₁, , Un} 1) Prove that if B is a basis for V, then Range(T) = span({T(u₁), .....,‚T(un)})

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Let \( V \) and \( W \) be vector spaces, and let \( L: V \rightarrow W \) be a linear transformation.

Let \( B = \{ u_1, \ldots, u_n \} \).

Prove that if \( B \) is a basis for \( V \), then

\[
\text{Range}(T) = \text{span}(\{ T(u_1), \ldots, T(u_n) \})
\]
Transcribed Image Text:Let \( V \) and \( W \) be vector spaces, and let \( L: V \rightarrow W \) be a linear transformation. Let \( B = \{ u_1, \ldots, u_n \} \). Prove that if \( B \) is a basis for \( V \), then \[ \text{Range}(T) = \text{span}(\{ T(u_1), \ldots, T(u_n) \}) \]
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