range and null space of T

by cocobaby
Tags: null, range, space
cocobaby is offline
Mar10-10, 09:04 AM
P: 9
Given a linear transformation T from V to V, can we say that the range of T is in the space spanned by the column vectors of T. And we already know that the null space of T is the one spanned by the set of vectors that are orthogonal to the row vectors of T, then is there any general relationship b/t the range of T and the nulll space of T ?
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HallsofIvy is offline
Mar10-10, 10:49 AM
Sci Advisor
PF Gold
P: 38,879
Yes, the "rank-nullity" theorem: If T is a linear transformation from U to V then the nulliity of T (the dimension of the null space of T) plus the rank of T (the dimension of the range of T in V) is equal to the dimension of U.

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