課程名稱︰線性代數
課程性質︰必修
課程教師︰呂學一
開課學院：電機資訊學院
開課系所︰資訊工程學系
考試日期（年月日）︰103/12/16
考試時限（分鐘）：60分鐘
試題 :
台大資工雙班線性代數第三次小考
2014年12月16日下午四點起一個小時
總共四題，每題十分，可按任何順序答題
第一題 Prove that
det(AB) = det(A)det(B)
holds for any A, B ∈ M_nxn(F). You may use the fact the above equality holds
when B is an elementary matrix. You may also use any properties of elementary
matrices shown in class including the 解剖定理, which states that an invertible
matrix is the product of a finite sequence of elementary matrices. However, you
have to prove the statement, if needed by your proof, that each non-invertible
square matrix has determinant 0_F.
第二題 Find the solution set K(S) for the following system S of linear
equations:
╭ ╮
╭ ╮│ x │ ╭ ╮
│ 3 -1 1 -1 2 ││ 1│ │ 5 │
│ ││ x │ │ │
│ 1 -1 -1 -2 -1 ││ 2│ │ 2 │
│ ││ x │ = │ │.
│ 5 -2 1 -3 3 ││ 3│ │10 │
│ ││ x │ │ │
│ 2 -1 0 -2 1 ││ 4│ │ 5 │
╰ ╯│ x │ ╰ ╯
╰ 5╯
第三題 State and prove Cramer's rule (克拉瑪公式)
第四題 Let
T_1 : U → V
T_2 : V → W
be linear, where U, V, and W are finite-dimensional vector spaces over F. Prove
rank(T_2T_1) ≦ min{rank(T_2), rank(T_1)}.
You may assume dim(T(X)) ≦ dim(X) holds for any linear T : X → Y, where X and
Y are finite-dimensional vector spaces over F.