let be z a fixed angle . if x belong to R2, let L(x)be the element.pdf

A

let be z a fixed angle . if x belong to R2, let L(x)be the element of R2 obtained by rotating x thru the angle z . assume L:R-R is linear . let u1=e1, u2=e2. then u1,u2 form a basisfor R2 . find a formula for L(x) j hint: first determine v1=L(u1) and v2=L(u2)k let be z a fixed angle . if x belong to R2, let L(x)be the element of R2 obtained by rotating x thru the angle z . assume L:R-R is linear . let u1=e1, u2=e2. then u1,u2 form a basisfor R2 . find a formula for L(x) j hint: first determine v1=L(u1) and v2=L(u2)k Solution First consider the 2x2 matrix that rotates vectors in R^2 by an angle of z: \$$A = \\left( \\begin{array}{cc} cos(z) & -sin(z) \\\\ sin(z) & cos(z) \\end{array} \ ight) \$$ Let \$$x = [x_1, x_2]^T = x_1e_1 + x_2e_2 \$$. Then \$$Ax = \\left( \\begin{array} { c } cos(z) x_1 - sin(z) x_2 \\\\ sin(z) x_1 + cos(z) x_2 \\end{array} \ ight) \$$ Therefore we define linear transformation L: R^2 --> R^2 by \$$L(x) = (cos(z) x_1 - sin(z) x_2, sin(z) x_1 + cos(z) x_2 ) \$$ where \$$x = (x_1, x_2) \$$..

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let be z a fixed angle . if x belong to R2, let L(x)be the element.pdf

• 1. let be z a fixed angle . if x belong to R2, let L(x)be the element of R2 obtained by rotating x thru the angle z . assume L:R-R is linear . let u1=e1, u2=e2. then u1,u2 form a basisfor R2 . find a formula for L(x) j hint: first determine v1=L(u1) and v2=L(u2)k let be z a fixed angle . if x belong to R2, let L(x)be the element of R2 obtained by rotating x thru the angle z . assume L:R-R is linear . let u1=e1, u2=e2. then u1,u2 form a basisfor R2 . find a formula for L(x) j hint: first determine v1=L(u1) and v2=L(u2)k Solution First consider the 2x2 matrix that rotates vectors in R^2 by an angle of z: ( A = left( begin{array}{cc} cos(z) & -sin(z) sin(z) & cos(z) end{array} ight) ) Let ( x = [x_1, x_2]^T = x_1e_1 + x_2e_2 ). Then ( Ax = left( begin{array} { c } cos(z) x_1 - sin(z) x_2 sin(z) x_1 + cos(z) x_2 end{array} ight) ) Therefore we define linear transformation L: R^2 --> R^2 by ( L(x) = (cos(z) x_1 - sin(z) x_2, sin(z) x_1 + cos(z) x_2 ) )
• 2. where ( x = (x_1, x_2) ).
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