4. Let L : V → R5 be a linear transformation. a. If L is onto and dim(kerL) = 2, what is dimV? b. If L is one-to-one and onto, what is dimV?

2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: tr : R" +R a11 a12 a13 ain ... a22 a23 a2n ... a21 a32 a33 Ha11 + a22 + a33 + ...+ ann = ... a31 i=1 An3 ann ... anl an2 Let In be the n x n identity matrix (or equivalently, a vector in R""). Calculate (A) that tr(In) = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and bri. Then compute tr(AB) and tr(BA).

38. a. Consider a linear transformation T from R³ to R. What are the possible values of dim(ker T)? Explain. b. Consider a linear transformation T from R4 to R'. What are the possible values of dim(imT)? Explain.

1. Let T be a linear transformation from M₂,2 A+ A+ A¹. M₂2. defined by the map Show that T is a linear transformation. What is its kernel?

Prove that T(x1,x2,?3) = [x1,x2] is a linear transformation.

Let T: M₂2 P₂ be a linear transformation. Prove that T is not one-to-one.

7. Is there a linear transformation T : R³ → R³ such that T If so, what is its matrix?

SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +