Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V (hence dim V = 2). Consider u, uz € V, and define the linear map T : V → V by T(e1) = u1 T(e2) = u2. a) Show that...

Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V<br>(hence dim V = 2).<br>Consider u, uz € V, and define the linear map T : V → V by<br>T(e1) = u1<br>T(e2) = u2.<br>a) Show that u,, uz is a basis of V if and only if<br>|(u1, u2)| < ||us|||42||-<br>b) Show that T is one-to-one if and only if T is onto (in which case it is an isomor-<br>phism).<br>c) Show that T is an isomorphism if and only if |(u), u2)| < ||u,|| ||12||<br>d) Show that ||T(u)|| = ||u|| for all u E V if and only if ||u |ll=||u2|| = 1 and (u1, U2) = 0.<br>

Extracted text: Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V (hence dim V = 2). Consider u, uz € V, and define the linear map T : V → V by T(e1) = u1 T(e2) = u2. a) Show that u,, uz is a basis of V if and only if |(u1, u2)| < ||us|||42||-="" b)="" show="" that="" t="" is="" one-to-one="" if="" and="" only="" if="" t="" is="" onto="" (in="" which="" case="" it="" is="" an="" isomor-="" phism).="" c)="" show="" that="" t="" is="" an="" isomorphism="" if="" and="" only="" if="" |(u),="" u2)|="">< ||u,||="" ||12||="" d)="" show="" that="" ||t(u)||="||u||" for="" all="" u="" e="" v="" if="" and="" only="" if="" ||u="" |ll="||u2||" =="" 1="" and="" (u1,="" u2)="">

Jun 05, 2022

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Exercise 4. Suppose that e1, e2 is an orthonormal basis of the real inner product V (hence dim V = 2). Consider u, uz € V, and define the linear map T : V → V by T(e1) = u1 T(e2) = u2. a) Show that...

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