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domain range A A -1 Pamela Leutwyler

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A Square matrix with 1’s on the diagonal and 0’s elsewhere Is called an IDENTITY MATRIX. I For every vector v, I v = v

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A square matrix A has an inverse if there is a matrix A -1 such that: AA -1 = I

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Only one to one mappings can be inverted: v v R v R Is the counterclockwise Rotation of through degrees. v v v RIf you know the value of You can find because Rotation is 1 – 1 (invertible) v P Is the projection of onto w v w v

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Only one to one mappings can be inverted: v v R v R Is the counterclockwise Rotation of through degrees. v v v RIf you know the value of You can find because Rotation is 1 – 1 (invertible) v P Is the projection of onto w v w v P is NOT 1-1. Given P v, v could be any one of many vectors vvv P is NOT invertible

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Now we will develop an algorithm to find the inverse for a matrix that represents an invertible mapping.

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A A -1 = I To solve for a, b, c, reduce: To solve for d, e, f, reduce: To solve for g, h, j, reduce:

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To solve for a, b, c, reduce: To solve for d, e, f, reduce: To solve for g, h, j, reduce: It is more efficient to do the three problems below in one step

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It is more efficient to do the three problems below in one step 1 1 0

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It is more efficient to do the three problems below in one step -2 0 1 3

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It is more efficient to do the three problems below in one step -4 7 0

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It is more efficient to do the three problems below in one step 3 0 -8 4

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A I I A -1 reduces to:

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1 1.8 © 2016 Pearson Education, Inc. Linear Equations in Linear Algebra INTRODUCTION TO LINEAR TRANSFORMATIONS.

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