1+i i 2i1 -i 2 C: An example of diagonalisation : Let T be the operator of C³ with matrix M = ( 7 0 1) Find 3 vectors e₁ e2 e3 satisfying the equations : T(e₁) = e₁ T(e₂) = ie2 T(e3-ie3

Let a € R, and define vectors u₁, U₂, U3 € R³ in terms of a by - 8. a U₁ = 2 U₂ = 3a + 3 2a +3 a² + 3a 7 [4a - 2] 13 = 3a-1 2a² (a) Show that the rank of the matrix [u₁ u₂ u3] does not depend on a. Hint: Find the rank by putting the matrix in row-echelon form. (b) Making reference to a fact from the course, show that u₁, U₂, U3 never span R³, no matter what value a takes. (c) Express u3 as a linear combination of u₁ and u2, i.e., as c₁u₁ + c₂U2. The scalars c₁ and c₂ will depend on a.

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Let A be a 3x3 matrix such that Assume that the vector V = E -2 is a solution of the matrix equation 3 Enter the vector V₁ in the form [C₁, C2, C3]: -12 12 -24 Find three vectors V₁, V₂, V3, different from V, which are also solutions of this equation. Enter the vector V₂ in the form [C1, C2, C3]: Nul (A) = Span (3) ED Enter the vector V3 in the form [C1, C2, C3]: Ax=