This means the lth column of a is replaced by a

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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.

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