推 phage17:請補填授課教師 04/25 23:19
※ 編輯: chengfred 來自: 140.112.242.90 (04/26 18:16)
課程名稱︰線性代數
課程性質︰系定必修(馮蟻剛、馮世邁、黃升龍、劉志文)
課程教師︰電機系統一教學
開課學院：電機學院
開課系所︰電機系
考試日期（年月日）︰96/4/25
考試時限（分鐘）：100
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
3 3 3
1. Suppose that T:R →R is the reflection of R about the x-y plane.
(a) What is the null space of T? (5%)
(b) Is T one-to-one? (5%)
(c) Is T onto? (5%)
(d) What is the standard matrix of T? (5%)
2. Show that if S is linearly dependent, then every vector in the span of S can
be written as a linear combination of the vectors in S in more than one way.
(15%)
3. Let A be an n×n matrix. Show that det(-A)=-det(A) if and only if n is odd.
(15%)
4. For any two subspaces V and U of Rn, the set W ={ w | w = v + u, where v
belongs to V and u belongs to U} and the set Z = V ∩ U are known to be
subspaces of Rn. Suppose Z ≠ { 0 } has a basis Bz = { z1, …, zk }, V has a
basis Bv ={ z1, …, zk, vl, …, zp } obtained from Bz using the extension
principle, and U has a basis Bu ={ z1, …, zk, ul, …, uq } obtained from Bz
using the extension principle.
(a) Prove that the dim(W) = dim(V) + dim(U) - dim(Z), where dim(‧) is the
dimension of the argument subspace. (10%) (Hint: Establish a basis for W.)
(b) For V = Null(A) and U = Null(B) with the following matrices A and B, find
a basis for Z. (5%)
(c) For V and U in (b), determine dim(V) + dim(U), and dim(Z) without using
the formula in (a), Verify the formula given in (a) with your results. (10%)
＿ ＿ ＿ ＿
∣ -9 -6 -10 -52 ∣ ∣ 1 1 0 1 ∣
A = ∣ -3 -6 5 20 ∣, B = ∣ 0 1 2 7 ∣
∣ 6 2 -1 6 ∣ ¯ ¯
¯ ¯ T T
5. Given a subspace N = Span{ [2 0 -1] , [-3 1 4] }, let F : V → V be a
linear function (i.e., F(au+bv)= aF(u) + bF(v) for all a, b belong to R and u,
T T T T
v belong to V). Suppose F([2 0 -1] ) = [-1 1 3] and F([-3 1 4] ) = [5 -1 -5] .
T
(a) F([2 0 -1] ) = ? (5%)
3 3
(b) Find a 3×3 matrix A such that the matrix transformation TA: R →R induced
by A has the property that TA(v) = F(v) for all v belongs to V. (5%)
(c) Is the matrix A you find in (b) unique? Why or why not? (5%)
(d) Is F onto as a function from V to V? One-to-one? Explain your answers. (10%)
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