課程名稱︰工程數學-線性代數 課程性質︰電機系大一必修 課程教師︰馮世邁 開課學院：電機資訊學院 開課系所︰電機工程學系 考試日期（年月日）︰2015/05/28 考試時限（分鐘）：50分鐘 試題 : 1.Let A be the matrix defined by [ 2 1 1 ] A = [ 1 2 1 ] [ 1 1 2 ] (a)(10%) Find the characteristic polynomial (CP) of A. (b)(15%) Find a basis for each eigenspace of A. (c)( 5%) Find an invertible P and a diagonal D such that A=PDP^(-1). (d)( 5%) Find a matrix B such that B^3=A. 2.Let A be an 3x3 matrix with det(A)=-2. Let the characteristic polynomial (CP) of A be f(t)= p3*t^3 + p2*t^2 + p1*t + p0. (a)( 3%) What is the value of p3 ? (b)( 4%) What is the value of p0 ? (c)( 4%) What is the CP of A^T ? (d)( 4%) What is the CP of -A ? (For the part (c)(d), you may express your answer in terms of p0, p1, p2, p3.)(No explanation is needed.) 3.Let W = Col C where C and the vectors u and v are respectively defined by [ 1 2 1 ] [ 1 ] [ 1 ] C = [ -1 -1 -1 ] , u = [ 0 ] , v = [ 1 ] [ 0 0 1 ] [ 1 ] [ 1 ] [ 1 3 1 ] [ 2 ] [ 1 ] (a)(14+6%) Find an orthogonal basis and an orthonormal basis for W. (b)( 5%) The vector u is in W. Write u as a linear combination of the vectors in your orthonomal basis for W in part(a). (c)(10%) Find a vector w 屬於 W and a vector z 屬於 W(perp), such that v = w + z. (d)( 5%) Find P(Wperp). (Hint dim W(perp)=1.) (e)( 5%) Find P(W). 4.(10%) Consider the nxn matrix uv^(T) where u and v are nx1 vectors that satisfy v^(T)u = 1. Prove that uv^(T) is diagonalizable. --

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