Unformatted text preview: u2,n a2 + · · · + un,n an ] .
This means the lth column of A is replaced by a linear combination of the ﬁrst l columns.
3. What if we multiply a matrix on the left by a diagonal matrix? The rows of the matrix are
multiplied by the corresponding diagonal entries.
4. What if we multiply a matrix on the left by an upper triangular matrix? The lth row of the
matrix is replaced by a linear combination of rows with indices ≥ l. 1 1 Review 1.1 Range and nullspace 1.1.1 Deﬁnition. We write Mm,n ≡ Mm,n (C) for the space of all complex m × n matrices. We
identify A ∈ Mm,n and the map A : Cn → Cm , x → Ax. Also, we abbreviate Mn ≡ Mn,n .
Similarly, we identify the real m × n matrices Mm,n (R) with maps from Rn to Rm . For A ∈ Mm,n ,
we let
ranA := {Ax : x ∈ Cn }, kerA = {x ∈ Cm : Ax = 0} .
These subspaces have dimensions rkA := dim ranA, nulA = dim kerA .
We recall a result from linear algebra.
1.1.2 Proposition. For A ∈ Mm,n , rkA + nulA = n.
Proof. Choose a basis for the subspace kerA, then complement...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.
 Fall '12
 BernhardBodmann
 Math, Matrices

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