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Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations...

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Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a

Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 → R2 be a linear transformatio! 1. State the Dimension Theorem for T. 2. Show that T is not 1-1. 3. Give an example for which T is onto. Give an example for which T is not onto. (In each case show that your example has the required property. Do not just give an example with no explanation). Problem 6, Let T : V → W be a linear map. Suppose that it is one-to-one. Suppose th be the images . . . . ,蚴is a linearly independent subset of V. Let wi 2], . . . ,we-Tuk. Show that w].. that { uh are linearly independent.
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un de benden ou Q. neng independen , 一 Ais indebendenti rn undebenden ve liunea indda

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Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations...
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