8. Linear transformation on R2 consists first on flipping the y-y and then counterclockwide rotation by 45°. Write the matrix of the transformation.

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Cross product is a linear transformation from R³ →R³ You can define it as X1 X2 X3 X Yı Y2 Y3 e₁ X1 Y1 x2 Y2 Y3 = = det( e2 → ez X3 Note that this is the determinant of a matrix that has vectors as some of the entries. You can look at section 3.6 in the textbook for alternate definitions. Define the linear transformation Tas T(x) = x x 2 3 a. Compute T(ei), T(₂), Te) b. Use your answer from a. to write the standard matrix of the linear transformation T c. Use your answer from b. to compute T( 4 H 3

Let T : R2 → R² be the linear transformation which rotates vectors counter-clockwise about the origin by 7T/3 radians (60 degrees). (i) Find the standard matrix of the linear transformation T. (ii) Use your matrix to rotate the vector v = counterclockwise about the origin by 4 T/3 radians.