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）︰103/12/18
考試時限（分鐘）：60分鐘
試題 :
台大資工單班線性代數第三次小考
2014年12月18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 Recall that if A ∈ M_nxn(R) and a_i,j with 1 ≦ i,j ≦ n is the
element of A in the i-th row and the j-column, then
n
det(A) = Σ sgn(σ) Π a ,
σ∈S_n i=1 i,σ(i)
where sgn(σ) ∈ {1, -1} is the signature of permutation σ. Prove the
following Laplace expansion formula along the n-th row of A:
n n+j ~
det(A) = Σ (-1) a det(A ),
j=1 n,j n,j
~
where A_n,j is the submatrix of A obtained by deleting the n-th row and the
j-column.
第二題 Find the solution set K(S) for the following system S of linear
equations:
╭ ╮
╭ ╮ │ x │ ╭ ╮
│ 2 -3 6 9 4 │ │ 1│ │-5 │
│ │ │ x │ │ │
│ 3 -1 2 4 1 │ │ 2│ │ 2 │
│ │ │ x │ = │ │.
│ 1 -1 2 3 1 │ │ 3│ │ 1 │
│ │ │ x │ │ │
│ 7 -2 4 8 1 │ │ 4│ │ 6 │
╰ ╯ │ x │ ╰ ╯
╰ 5╯
第三題 Prove that a system S of linear equations has solutions if and only if
rank(A) = rank(B),
where A is the coefficient matrix of S and B is the augmented coefficient
matrix of S.
第四題 Prove that if A ∈ M_nxn(R), then A is invertible if and only if
rank(A) = n.