93年 - 93 淡江大學轉學考線性代數#56047

科目：轉學考-線性代數 | 年份：93年 | 選擇題數：0 | 申論題數：10

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所屬科目：轉學考-線性代數

選擇題 (0)

申論題 (10)

(a)Find the eigenvalues and eigenvectors*

(b) What is the answer of S_1MS, where S is the matrix whose columns are the eigenvectors of M

【已刪除】(a) Find a basis for the nullspace N(A) and a basis for the rowspace

【已刪除】(b) Show that nullspace N(X) and the rowspace

are orthogonal.

(c) Prove that the nullspace and the rowspace of any matrix are orthogonal.

【已刪除】 (a) Show that

【已刪除】(b) Prove that A =

【已刪除】4. 20% For the matrix

We can find a nonsingular matrix M such that A is similar to J, where J = M^-1AM and J is in Jordan form. What is the matrix J and M ? (Hint: Solve AM = MJ for columns of M. You can find two columns of M in the columns of A.)

【已刪除】 (a) Find the solution x such that Ax is the projection of b onto

. (Hint： Differentiate (b - Ax)^T(b - Ax) with respect to the vector x to have an equation.)

【已刪除】(b) The projection Ax in (a) can be represented as H6 where H is a m x m projection matrix, argue that H² = H, and then show the eigenvalues of H are 0’s or Vs. (Hint: The projection of Ax onto

is itself.)