課程名稱︰線性代數一
課程性質︰數學系必修
課程教師：陳其誠
開課學院：理學院
開課系所︰數學系
考試日期︰2006年12月06日
考試時限：未知
是否需發放獎勵金：是
試題 :
In this exam, define two linear spaces P and Pd, a 4x4-matrix A and ordered
basis B of R^4 as follow:
P={a0+a1x+...+anx^n}
and for integer d,
Pd={a0+a1x+...+adx^d}
also
┌ 1 2 -3 0 ┐
│ 3 5 2 -2 │
A = │-2 -3 -4 1 │
└ 1 0 1 0 ┘
and
B={v1=(1,1,1,1),v2=(1,0,1,-1),v3=(0,1,0,1),v4=(0,0,1,-1)}
(1) (a) Find the coordinate of the vector v =(3,1,9,-7) relative to the ordered
basis B (20 points).
(b) Suppose that T:R^4->R^4 is the linear transformation so that A is the
matrix representation of T relative to B,B. Calculate the coordinate of
the vector T(v) relative to the standard basis (20 points).
(2) Find a basis of the subspace
sp(x^3,x^2+1,x^2+x-1,x^3-x^2-1,2x-4)
in P (15 points).
(3) Let V=sp(1,x,x^2)⊂P4 and let
V^⊥ = {g∈p4│(f,g)=0,any f∈V},
where
1
(f,g)=∫ f(x)g(x)dx.
-1
Find a basis of V^⊥ (20 points).
(4) Find two subspace V and W in R^6 such that dimV=dimW=4 and
dimV∩W=2 (10 points).
(5) Let W be a subspace of a finite dimensional inner-product space V. Let
W^⊥={v∈V│(v,w)=0 any w ∈ W}.
(a) Show that W∩W^⊥={0} (5 points).
(b) Show that dimW+dimW^⊥dimV and for every x ∈ V there
are unique w ∈W and v ∈W^⊥ such that x=w+v (5 points).
(c) Suppose V = R^5 (with the standard inner-product),
W=sp((1,1,1,0,1),(,1,2,3,1,1))
and x=(1,0,0,0,0). Find the vectors w and v in (b) (5 points).
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.243.132