The linear transformation T: R2 R? with matrix cos 0 - sin 0 A = cos 0 rotates vectors in the ry-plane sin 0 counterclockwise through an angle 0, where 0 <0 < 2n. By arguing geometrically, determine all values of 0 for which A has real eigenvalues. Find the real eigenvalues and the corresponding eigenvectors.

If wis an eigenvector of A, how does the vector Au compare geometrically to the vector ? O A is a reflection of in the z-axis O A is a reflection of win the y-axis O Aw is parallel to w but may have a different length O Aw is a rotation of

Matrix A has the eigenvalue 2 = 5 and eigenvector v. Find generalized eigenvector. 7 1 01 1|,λ5, ν = A = -3 4 -2 1 41 X2=

Suppose A + u are distinct eigenvalues of A, with corresponding eigenvectors u and v. (Au)™v = Au" v O pu"v V O du7v (Check ALL that apply) uv = are column vectors corresponding to the vectors u, v, then u'v is a Prove your result. Suggestion: If u, matrix whose entry is i · v.