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Mathematics
Linear and Abstract Algebra
Proving Isometries: A Step-By-Step Guide
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[QUOTE="RUber, post: 5503839, member: 524408"] I see. thanks for the explanation. For the first one, assume there are two linear maps then show that they must be equal. Because a linear map can be uniquely defined by its matrix representation, showing that the matrix representation must be the same should work. ##D_{BB} ## is a matrix that takes an input from basis B and gives an output in basis B. ##D_{SS} ## is a matrix that takes an input from basis S and gives an output in basis S. Then, ##D_{BS} ## should be a matrix that takes an input from basis B and gives an output in basis S. Look at a simple example, Let ##V = \mathbb{R}^3 ##, then ##S = \{ \hat x, \hat y, \hat z\}##, ##W## is the xy-plane. ##W^\perp## is span of ##\hat z##. The matrix representation of ##[\Phi ]_{SS} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}## In your question 2, you are asked to give the matrix representation in the eigenbasis B...which should be pretty similar. [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Proving Isometries: A Step-By-Step Guide
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