notes-orthog

notes-orthog - Math 1b Prac Rowspaces nullspaces orthogonal...

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Math 1b Prac — Rowspaces, nullspaces, orthogonal spaces January 23, 2009 [As usual, the notes I post on the web are sketchy and more of an outline. Some more details in class.] Subspaces of R n arise, as an oversimpliFcation, in two ways: either as the span of some vectors or as the solutions of a homogeneous system of linear equations. It is useful to be able to go from one kind of description to the other. Given a matrix M ,the row space of M is the span of its rows (the set of all linear combinations of its rows). The nullspace (or solution space) of M is the set of all column vectors x so that M x = 0 . We will check (it is not hard) that these are really subspaces. Row operations on a matrix M do not change the row space or the nullspace of M . Given a subspace U of R n orthogonal space U (read: U perp)istheseto fa l l vectors y so that x · y =0fora l l x U . To check this condition ( x · y l l x U ), it is sufficient to check that y is orthogonal to a spanning set of U , so assuming we have a spanning set u 1 , u 2 ,..., u k for U , a vector y
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notes-orthog - Math 1b Prac Rowspaces nullspaces orthogonal...

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