課程名稱︰ 線性代數
課程性質︰ 系必修
課程教師︰ 陳文進
開課學院： 電資學院
開課系所︰ 資訊工程學系
考試日期（年月日）︰ 96/11/20
考試時限（分鐘）： 170 mins
是否需發放獎勵金： 是
試題 :
1.(15%) The equation 2X + 2Y - 3Z + 8W = 6 defines a hyperplane in R^4.
a. Give its normal vector α.
b. Find its distance from the origin using dot products.
c. Find the point on the hyperplane closest to the origin by using
the parametric equation of the line through 0 with direction
vector α. Double-check your answer in b.
d. Find the distance from the point ω = ( 1,1,1,1 ) to the hyperplane
using dot product.
e. Find the point on the hyperplane closest to ω by using the
parametric equations od the line through ω with direction vector
α. Double-check your answer in d.
2.(10%) Solve the system of equations:
︴λX + Y + Z = 1
︴ X + λY + Z = 1
︴ X + Y + λZ = 1
Note that the equations may be inconsistent, have unique solution, or have
infinite solution. You have to discuss the value of λ for each case.
3.(15%) ┌ ┐ ┌ ┐
│ 1 0 0 0 │ │ 1 0 0 0 │
Let matrices A = │-2 3 0 0 │, I= │ 0 1 0 0 │ and
│ 0 -4 5 0 │ │ 0 0 1 0 │
│ 0 0 -6 7 │ │ 0 0 0 1 │
└ ┘ └ ┘
B = (I+A)^(-1)(I-A), caculate the matrix (I+B)^(-1).
┌ ┐
4.(15%) Find the rank of the matrix A = │ 1 α -1 2 │
│ 2 -1 α 5 │
│ 1 10 -6 1 │
└ ┘
5.(15%) Suppose U and V are subspaces of R^n.
Prove that ( U + V )⊥ = U⊥ ∩ V⊥.
6.(15%) Let A be a n × n matrix and suppose ν1 ,ν2 ,ν3 ε R^n are nonzero
vectors that satisfy
Aν1 = ν1
Aν2 = 2ν2
Aν3 = 3ν3
Prove that { ν1 ,ν2 } must be linearly independent.
7.(15%) Using the method described in the textbook to find bases for R(A),
N(A), C(A), N(A⊥), where
┌ ┐
│ 1 1 0 5 0 -1 │
A= │ 0 1 1 3 -2 0 │
│-1 2 3 4 1 -6 │
│ 0 4 4 12 -1 -7 │
└ ┘
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