By Prabhat Choudhary
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Extra info for A Practical Approach to Linear Algebra
An important corollary of the rank theorem, is the following theorem connecting existence and uniqueness for linear equations. Theorem. Let A be an an m x n matrix. Then the equation Ax = b has a solution for every b ATx=O E lR m if and only if the dual equation 46 Systems of Linear Equations has a unique (only the trivial) solution. (Note, that in the second equation we have AT, not A). Proof The proof follows immediately from Theorem by counting the dimensions. There is a very nice geometric interpretation of the second rank theorem.
A reasonable theory can be derived from the following definition. Definition. Suppose that the two matrices a~ 1 a~n 1 .. [b~ I A=: : and B = : [ amI amn bm1 both have m rows and n columns. Then we write all ~ql al amI + bml amn ·+ bmn A+B=[ n .. b~:n 1 bmn ~bln 1 and call this the sum of the two matrices A and B. Example. Suppose that A=[ ~ ~11 ~ ; -1 0 7 and B 6 =[ ~ ~ ~2 ~Il' -2 1 3 3 Then A+B = [~:~ ;:~ ;~! -21~171 [~ ~ ~ 9~1' and -1-1 0+1 7+3 6+3 -1 1 10 Example. We do not have a definition for adding" the matrices 2 [ -1 4 3 0 7 -I] [~ ~l 6 and ; -1 0 7 Proposition.
After the backward phase of the row reduction, we get what the socalled reduced echelonform of the matrix: coefficient matrix equal I, as in the above example, is a particular case of the reduced echelon form. The general definition is as follows: we say that a matrix is in the reduced echelon form, if it is in the echelon form and 3. All pivot entries are equal I; 4. All entries above the pivots are O. Note, that all entries below the pivots are also o because of the echelon form. To get reduced echelon form from echelon form, we work from the bottom to the top and from the right to the left, using row replacement to kill all entries above the pivots.