課程名稱︰線性代數
課程性質︰大二上必修
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）︰2016/1/10
考試時限（分鐘）：180
試題 :
前十一題都是可能對也可能不對的敘述。如果你覺得對請證明它是對的，
如果你覺得不對請證明它是錯的。最後一題請按題目指示作答。
1.
Let V be a subspace of inner-product space W on |R with dim(W) < ∞.
If T is the orthogonal projection of W on V, then I - T* is the
W
⊥
orthogonal projection of W on V .
2.
Let T ∈ |L(V, V) for inner-product space V on |C with dim(V) < ∞.
-1
If T is self-adjoint, then (T + iI )(T - iI ) is unitary.
V V
3.
Let T ∈ |L(V, V) for inner-product space V on |R with dim(V) < ∞.
If T is normal and the characteristic polynomial of T splits, then
T is self-adjoint.
4.
Let V be an inner-product space with dim(V) < ∞. Let x, z ∈ V.
def
If T ∈ |L(V, V) is defined as T(y) === <x|y>z for any y ∈ V,
then T* does not exist.
5.
If U and V are subspaces of inner-product space W with dim(W) < ∞,
⊥ ⊥ ⊥
then (U ∩ V) = U + V .
6.
If d and d' are inner-product functions of a vector space V on |R,
then so is ad + d' for any positive real a.
7.
Let T ∈ |L(V, V) for vector space V with dim(V) < ∞. If T is
-1
invertible and diagonalizable, then T is diagonalizable.
8. n ×n n ×n
If B ∈ |C is invertible, then there is an A ∈ |C such that
A + cB is invertible for each c ∈ |C.
9. n ×n n ~
For any A ∈ |R with n ≧ 2, det(A) = Σ A det(A ).
i=1 i,i i,i
10. 3 ×3 -1
There is a Q ∈ |R such that Q AQ is diagonal, where
╭ 7 8 6 ╮
A = │-4 -5 -6 │.
╰ 0 0 3 ╯
11. 3 ×3
There is a unitary or orthogonal Q ∈ |C such that QAQ* is diagonal,
╭ 2 1 1 ╮
where A = │ 1 2 1 │.
╰ 1 1 2 ╯
12. 1 ×4
Let subset S of |R consist of the following four vectors:
(1, -2, -1, 3),
(3, 6, 3, -1),
(-1, 2, 1, 1),
(1, 4, 2, 8).
Find an orthonormal basis of span(S).