课程名称︰线性代数
课程性质︰系必修
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）︰2017/12/19
考试时限（分钟）：60
试题 :
共五题每题廿分，可按任何顺序答题。每题难度不同，请审慎判断恰当的解题顺序。
第一题
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.
第二题
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
第三题
(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.
第四题
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 .
第五题
Give the inverse of
-24 18 5
( 20 -15 -4 )
-5 4 1
无需计算过程。