課程名稱︰線性代數
課程性質︰系必修
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）︰2017/12/19
考試時限（分鐘）：60
試題 :
共五題每題廿分，可按任何順序答題。每題難度不同，請審慎判斷恰當的解題順序。
第一題
Let V and W be vector spaces over a scalar field F with dim(V) = n and
dim(W) = m. Give a basis for L(V,W). Justify your answer.
第二題
m x n n x m
Let A ∈ F and B ∈ F for positive integers m and n and a scalar
field F. Prove or disprove that A x B = I implies B x A = I .
m n
第三題
(1) (10 points) Prove or disprove that a function TR admits a left inverse
(i.e., a function L with LT = I) if and only if T is surjective.
(2) (10 points) Prove or disprove that a function T admits a right inverse
(i.e., a function R with TR = I) if and only if T is injective.
第四題
n m
Let T : F → F . Prove that
T is linear
m x n
if and only if there is a unique matrix A ∈ F such that
T(x) = Ax
n
holds for each x ∈ F .
第五題
Give the inverse of
-24 18 5
( 20 -15 -4 )
-5 4 1
無需計算過程。