课程名称︰线性代数
课程性质︰必修
课程教师︰吕学一
开课学院：电机资讯学院
开课系所︰资讯工程学系
考试日期（年月日）︰103/12/16
考试时限（分钟）：60分钟
试题 :
台大资工双班线性代数第三次小考
2014年12月16日下午四点起一个小时
总共四题，每题十分，可按任何顺序答题
第一题 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.
第二题 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╯
第三题 State and prove Cramer's rule (克拉玛公式)
第四题 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.