Stitz-Zeager_College_Algebra_e-book

A the adult male sasquatch needs 3 servings of

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Unformatted text preview: 2 = 3 · 2. We know from algebra4 that 3x = x + x + x, so it seems natural that given a matrix A, we define 3A = A + A + A. If A = [aij ]m×n , we have 3A = A + A + A = [aij ]m×n + [aij ]m×n + [aij ]m×n = [aij + aij + aij ]m×n = [3aij ]m×n In other words, multiplying the matrix in this fashion by 3 is the same as multiplying each entry by 3. This leads us to the following definition. Definition 8.8. Scalara Multiplication: We define the product of a real number and a matrix to be the matrix obtained by multiplying each of its entries by said real number. More specifically, if k is a real number and A = [aij ]m×n , we define kA = k [aij ]m×n = [kaij ]m×n a The word ‘scalar’ here refers to real numbers. ‘Scalar multiplication’ in this context means we are multiplying a matrix by a real number (a scalar). One may well wonder why the word ‘scalar’ is used for ‘real number.’ It has everything to do with ‘scaling’ factors.5 A point P (x, y ) in the plane can be represented by its position matrix, P : (x, y ) ↔ P = x y Suppose we take the point (−2, 1) and multiply its position matrix by 3. We have...
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