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Graphs and Adjacency Matrices:(see MATLAB Project # 2)A graph is a set of points (called vertices or nodes)
and a set of lines (called edges or paths of length one) connecting some pairs of nodes.
Two nodes connected by an edge are said to be adjacent.
The Adjacency Matrix for a graph with nodes is an n x n matrix where
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4.7 Change of BasisQuestion. Suppose that and are two bases for
a vector space Let be a vector in How are and related?
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4.7 Change of BasisQuestion. Suppose that and are two bases for
a vector space Let be a vector in How are and related?
Answer:
where is the change-of-coordinates matrix from to
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Theorem. Let be two bases for a vector space Then there is a unique
matrix such that
is called the change-of-coordinates matrix from to
Furthermore,
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Definition. An eigenvector of a square matrix is a nonzero vector such that for some scalarA scalar is called an eigenvalue of if there is a nontrivial solution of such an is called an eigenvector of corresponding to
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5.2 The Characteristic EquationRecall: eigenvector of corresponding to
nonzero.
How do we find the eigenvalues of