Let T: C′[0, 1]→C[0, 1] be the linear transformation T( f ) = f ′. Show that λ = 1 is an eigenvalue of T with corresponding eigenvector f (x) = ex.

Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

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Determine the eigenvalues and eigenvectors of the linear = (3x + 7y, 5x – 2y). map T : R² → R² defined by T(x, y) -

Consider a linear transformation T from R³ to R² for which T ¹ (C) - · ~ (C) -· ¹(E) -- T = T Find the matrix A of T. = B

Let L be the line in R³ spanned by the vector 2. Let A be the linear transformation given by rotating 180° rotation around the axis L. What are the eigenvalues of A? Justify your answer.

7. Let w E R³ be a fixed vector with ||w|| = 1 and let f : R³ → R³ be the linear transformation given by f(v) = v – 2(v . w)w. | Let M be the standard matrix of f. Find two eigenvalues of M and show that M is diago- nalizable.

T is a linear operator and T: (a,b) → (2a,4b) a) Explain the geometric meaning of the eigenvalues. b) Explain geometrically the Eigen spaces of the linear transformation

3- Consider the square of the derivative operator D2. a) Show that D² is a linear operator. b) What is the eigenfunctions and corresponding eigenvalues of D2. c) Give an example of an eigenfunction of D² which is not an eigenfunction of D.