2331_Notes_3o6_fill

2331_Notes_3o6_fill - Math 2331 Linear Algebra Section 3.6...

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Math 2331 Linear Algebra Section 3.6 The Fundamental Subspaces of a Matrix Matrix A is M rows by N columns The fundamental subspaces associated with A are - 1. The column space of A C(A) or Col (A) all linear combinations of the column vectors of A. 2. The null space of A - all solutions to the equation Ax = 0 and denoted N(A) 3. The Row space of A - all linear combinations of the row vectors of A. ALSO - this is the COLUMN SPACE of the transpose of A. ( ) T CA 4. The left nullspace of A - The NULL SPACE of the transpose of A - ( ) T NA Let's look at these one at a time, both for a row-reduced echelon form matrix and for a general M x N matrix. COLUMN SPACE Remember that A has m rows and n columns, so we can think of A as n column vectors. HOWEVER, How many elements are in each column vector? So, the column space lives in M-dimensional space (the number of rows!!!!!!) The column space is the span of all the column vectors. All of them. However, some of the columns already may be linear combinations of the other columns so one of the questions to ask is which ones do you really need to generate the column space. The answer is that you only need the pivot columns. The other columns are combos of the pivot columns.
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2331_Notes_3o6_fill - Math 2331 Linear Algebra Section 3.6...

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