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).

Please don't provide handwritten solution ....

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

Define the linear map T : P3(R) → P3(R) by T(p(x)) = xp'(x) – p(x). - (а) Determine the eigenvalues and eigenvectors. (b) Show that P3(R) = Ker(T) Im(T).

The picture below shows a vector v in R2 and its image under a linear transformation T:R2 → R2. Explain why v is not an eigenvector of the linear transformation T. X2 axis V I1 аxis T(v)

Explain.

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

The linear transformation T : R2 → R2 with matrixA = cos θ − sin θsin θ cos θ rotates vectors in the x y-plane counterclockwise through an angle θ, where 0 ≤ θ <2π. By arguing geometrically, determine all valuesof θ for which A has real eigenvalues. Find the realeigenvalues and the corresponding eigenvectors.