Engineering Calculus Notes 434

Engineering Calculus Notes 434 - x perpendicular to the...

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422 CHAPTER 4. MAPPINGS AND TRANSFORMATIONS (o) f ( x,y,z ) = ( x + y, 2 x y, 3 x + 2 y ) (p) f ( x,y,z ) = ( x + y + z,x 2 y + 3 z ) (q) Projection of R 3 onto the plane 2 x y + 3 z = 0 (r) Projection of R 3 onto the plane 3 x + 2 y + z = 1 (s) Rotation of R 3 around the z -axis by θ radians counterclockwise, seen from above. (t) Projection of R 3 onto the line x = y = z . 2. Express each aFne map T below as T ( −→ x ) = T ( −→ x 0 ) + L ( −→ x ) with the given −→ x 0 and linear map L . (a) T ( x,y ) = ( x + y 1 ,x y + 2) , −→ x 0 = (1 , 2) (b) T ( x,y ) = (3 x 2 y + 2 ,x y ) , −→ x 0 = ( 2 , 1) (c) T ( x,y,z ) = (3 x 2 y + z,z + 2) , −→ x 0 = (1 , 1 , 1) (d) T ( x,y ) = (2 x y + 1 ,x 2 y, 2) , −→ x 0 = (1 , 1) (e) T ( x,y,z ) = ( x + 2 y,z y + 1 ,x ) , −→ x 0 = (2 , 1 , 1) (f) T ( x,y,z ) = ( x 2 y + 1 ,y z + 2 ,x y z ) , −→ x 0 = (1 , 1 , 2) (g) T ( x,y,z ) = ( x + 2 y z 2 , 2 x y + 1 ,z 2) , −→ x 0 = (1 , 1 , 2) (h) T is projection onto the line −→ p ( t ) = ( t,t + 1 ,t 1), −→ x 0 = (1 , 1 , 2) ( Hint: ±ind where the given line intersects the plane through
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Unformatted text preview: x perpendicular to the line.) (i) T is projection onto the plane x + y + z = 3, x = (1 , 1 , 2) ( Hint: rst project onto the parallel plane through the origin, then translate by a suitable normal vector.) Theory problems: 3. Prove Remark 4.1.1 . ( Hint: What is the coordinate column of the standard basis vector e j ?) 4. Show that the composition of two aFne maps is again aFne. 5. ind the matrix representative for each kind of linear map L : R 2 R 2 described below: (a) Horizontal Scaling : horizontal component gets scaled (multiplied) by > 0, vertical component is unchanged....
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