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Unformatted text preview: Lecture 2 Matrix Operations • transpose, sum & difference, scalar multiplication • matrix multiplication, matrixvector product • matrix inverse 2–1 Matrix transpose transpose of m × n matrix A , denoted A T or A ′ , is n × m matrix with ( A T ) ij = A ji rows and columns of A are transposed in A T example: 0 4 7 0 3 1 T = bracketleftbigg 0 7 3 4 0 1 bracketrightbigg . • transpose converts row vectors to column vectors, vice versa • ( A T ) T = A Matrix Operations 2–2 Matrix addition & subtraction if A and B are both m × n , we form A + B by adding corresponding entries example: 0 4 7 0 3 1 + 1 2 2 3 0 4 = 1 6 9 3 3 5 can add row or column vectors same way (but never to each other!) matrix subtraction is similar: bracketleftbigg 1 6 9 3 bracketrightbigg − I = bracketleftbigg 0 6 9 2 bracketrightbigg (here we had to figure out that I must be 2 × 2 ) Matrix Operations 2–3 Properties of matrix addition • commutative: A + B = B + A • associative: ( A + B )+ C = A +( B + C ) , so we can write as A + B + C • A + 0 = 0 + A = A ; A − A = 0 • ( A + B ) T = A T + B T Matrix Operations 2–4 Scalar multiplication we can multiply a number (a.k.a. scalar ) by a matrix by multiplying every entry of the matrix by the scalar this is denoted by juxtaposition or · , with the scalar on the left: ( − 2) 1 6 9 3 6 0...
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 Spring '10
 Cudeback
 Linear Algebra, Matrix Operations

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