Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T : V → W is linear, prove that (x, y), = (T (x),T (y)), defines an inner product on V if and only if T is one-to-one. %3D

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Let V be a vector space over R and W be an inner product space over R with inner
product (, ),. If T: V → W is linear, prove that (r, y), = (T (x),T (y)), defines an inner
product on V if and only if T is one-to-one.
Transcribed Image Text:Q5 Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T: V → W is linear, prove that (r, y), = (T (x),T (y)), defines an inner product on V if and only if T is one-to-one.
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