Engineering Calculus Notes 435

Engineering Calculus Notes 435 - ) are linearly dependent....

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4.1. LINEAR MAPPINGS 423 (b) Vertical Scaling : vertical component gets scaled (multiplied) by λ > 0, horizontal component is unchanged. (c) Horizontal Shear : Each horizontal line y = c is translated (horizontally) by an amount proportional to c . (d) Vertical Shear : Each vertical line x = c is translated (vertically) by an amount proportional to c . (e) Reflection about the Diagonal : x and y are interchanged. (f) Rotation : Each vector is rotated θ radians counterclockwise. Challenge problems: 6. Suppose L : R 2 R 2 is linear. (a) Show that the determinant of [ L ] is nonzero iF the image vectors L ( −→ ı ) and L ( −→ ) are independent. (b) Show that if L ( −→ ı ) and L ( −→ ) are linearly independent, then L is an onto map. (c) Show that if L ( −→ ı ) and L ( −→ ) are linearly dependent , then L maps R 2 into a line, and so is not onto. (d) Show that if L is not one-to-one, then there is a nonzero vector −→ x with L ( −→ x ) = −→ 0 . (e) Show that if L is not one-to-one, then L ( −→ ı ) and L ( −→
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Unformatted text preview: ) are linearly dependent. (f) Show that if L ( ) and L ( ) are dependent, then there is some nonzero vector sent to 0 by L . (g) Use this to prove that the following are equivalent: i. the determinant of [ L ] is nonzero; ii. L ( ) and L ( ) are linearly independent; iii. L is onto; iv. L is one-to-one. (h) L is invertible if there exists another map F : R 2 R 2 such that L ( F ( x,y )) = ( x,y ) = F ( L ( x,y )). Show that if F exists it must be linear. 7. Show that every invertible linear map L : R 2 R 2 can be expressed as a composition of the kinds of mappings described in Exercise 5 . ( Hint: Given the desired images L ( ) and L ( ), rst adjust the angle, then get the lengths right, and nally rotate into position.)...
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