Let L : R^2 → R^2 be a linear transformation. If L((1, 2)^T) = (2, −3)T and L((1, −1)^T) = (5, 2)^T, determine the values of L((7, 5)^T).

If T: R³ R³ is a linear transformation such that then T **((:))- -5 T (6D)-B- (E)-13 (ED-EL- = -3 T T "

Define the function Q : M2x2 M2x2 by Q(A) = A – AT. Show that Q is a linear transformation.

1). Corresponds to the Transformation k(x,y) = ( x + 3, y-5).  Fully describe the transformation K. 2). Corresponds to the Transformation H(x,y) = (0.5x, 0.5y). Fully describe the transformation H.

Suppose T : R^2 → R^2 is a linear transformation that does not alter e_i andwhich maps e_2 into e_2 + 2e_1, where e_i = (1,0), e_2 = (0,1). Find the transformation matrix T.Justify your answer.

6. (10) Is the transformation defined by T(x,х2) Explain. 3D (2х,, Зх, -5х2, —х, + 1) linear?

8. Is there a linear transformation T: R³ → R² such that T(1, 0, 3) = (1, 1) and T(−2, 0, −6) = (2, 1) ? Explain.

2 -2 -[:]~-~-[;) - - [³ ]-~^»-[*³] and and e₂= and Y₂ = 5 6 Let e, = into y₂. Find the images of The image of 5 [3] -3 is 5 5] -3 and X₂ X2 and let T. R² R² be a linear transformation that maps e, into y, and maps e₂ →

2 Let T₁: P₂ → R² and T₂: R² → R2x2 be linear transformations defined as follows. Ti (az² + bx + c) = [a+b] ¹([2])-[ (T₂0 T₁) (2x² + 4x + 5) = 5х1 2x1 -9x2 -8x2 [ Ex: 42

Define a linear transformation T: P₂ → R² by T (p) = [] p(0) ->> p(0) °‹[a] : a is a real number} O The basis of the range of T is {t, t²} The entire set of all real numbers, R [8] The entire set R² [3] : 0 is a real number} : a and b are real numbers} Describe the range of T.