課程名稱︰數值線性代數 課程性質︰數學系選修 課程教師︰薛克民 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2016/11/01 考試時限（分鐘）：110 試題 : Instructions: ‧ Total points 100 ‧ Open books, notes, and laptops ‧ Computer software can only be used for numerical validation, but is not used directly for the solution ‧ Answer the questions thoroughly and justify all your answers m ×m m T -1 1. (20 points) Let A ∈ |R be nonsingular, u, v ∈ |R , and v A u ≠ -1. Show that -1 T -1 T -1 -1 A uv A (A + uv ) = A - ─────. T -1 1 + v A u 2. (30 points) Consider the matrix ┌ 1 1 ┐ │ │ A = │ 0 0 │. │ ＿ ＿│ └ √2 -√2 ┘ Λ Suppose that we have known Σ and V* of the reduced singular value decomposition (SVD) of the matrix A as ＿ ＿ ^Λ ^┌ 2 0 ┐┌ √2/2 -√2/2 ┐ A = UΣV* = U│ ＿││ ＿ ＿ │. └ 0 √2┘└ √2/2 √2/2 ┘ ^ (a) Compute U. (b) Construct a full SVD of A = UΣV* for some U and Σ. (c) What are the four fundamental subspaces of A, i.e. R(A), N(A*), R(A*), and N(A)? (d) What is the pseudoinverse of A? m ×m 3. (25 points) Let Q = Q = [q , q , ..., q ] ∈ |R be a real orthogonal 1 1 2 m matrix. T (a) Determine a reflector P = I - 2u u such that P q = e . 1 1 1 1 1 1 (b) Show that P Q = Q has the form 1 1 2 ┌ 1 0 ... 0 ┐ │ │ │ 0 │ Q = │ ~ │ 2 │ : Q │ │ 2 │ └ 0 ┘ ~ ~ ~ ~ (m-1) ×(m-1) where Q = [q , q , ..., q ] ∈ |R is a real orthogonal 2 1 2 m-1 matrix. (c) Using the result in (b), Q can be transformed into a diagonal form with a sequence of orthogonal transformations P ...P P . m-1 2 1 What is this diagonal form? 4. (25 points) Consider the matrix ┌ 3 -3 ┐ │ │ A = │ 0 4 │. │ │ └ 4 1 ┘ (a) Find the QR factorization of A by Householder reflection. (b) Use the result in (a) to find the least squares solution of Ax = b, where T b = [16 11 17] . (c) What is the orthogonal projector of A into R(A)? -- 移居二次元(|R^2)的注意事項： 3. 如果你在從事random walk，往上下左右的 1. connectedness不保證pathwise connec- 的機率都是1/4，則你能回家的機率是1。 tedness。可能你跟你的幼馴染住很近， 4. 下面這個PDE是二次元上的波方程式 卻永遠沒辦法到她家。 http://i.imgur.com/2H9HllP.png 2. ODE的C^1 autonomous system不會出現 它的解不滿足Huygens' principle，因此 chaos，在預測事情上比較方便。 講話時會聽到自己的回音，很不方便。 --

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