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(a) Let T: VW be a linear transformation defined on vector spaces from V to W. If T(v₁),T(v₂),,T(v) are linearly independent vectors in W, show that V₁, V2,, Vp are linearly independent in V. (b) Let T be a linear transformation defined by T(x₁, x₂) = (2x₁ + x₂, -x₁ + x2,9x₂). i. Is T one-to-one? Justify your answers. ii. Is T onto? Justify your answers.

(a) Let T: VW be a linear transformation defined on vector spaces from V to W. If T(v₁),T(v₂),,T(v) are linearly independent vectors in W, show that V₁, V2,, Vp are linearly independent in V. (b) Let T be a linear transformation defined by T(x₁, x₂) = (2x₁ + x₂, -x₁ + x2,9x₂). i. Is T one-to-one? Justify your answers. ii. Is T onto? Justify your answers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(a) Let T: VW be a linear transformation defined on vector spaces from V to W. If T(v₁),T(v₂),...,T(v) are linearly independent vectors in W, show that V₁, V2,, Vp are linearly independent in V. (b) Let T be a linear transformation defined by T(x₁, x₂) = (2x₁ + x₂, -x₁ + x2,9x₂). i. Is T one-to-one? Justify your answers. ii. Is T onto? Justify your answers.
Transcribed Image Text:(a) Let T: VW be a linear transformation defined on vector spaces from V to W. If T(v₁),T(v₂),...,T(v) are linearly independent vectors in W, show that V₁, V2,, Vp are linearly independent in V. (b) Let T be a linear transformation defined by T(x₁, x₂) = (2x₁ + x₂, -x₁ + x2,9x₂). i. Is T one-to-one? Justify your answers. ii. Is T onto? Justify your answers.
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