108年 - 108 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數和離散數學)#106050

1. (5%) Consider the following undirected graph. Write down 4 nodes which form an independent set.

2. (5%) A palindrome is a sequence of symbols that reads the same left to right as right to left. What is the number of palindromic binary numbers of length n?

3. (5%) Derive
A =_______
B=________

4. (10%) Derive
A=_________
B=___________

5. (10%) The solution to the recurrence equation
in terms of (the arbitrary initial conditions) a₀and a₁.

6.(10%) Consider

x₁+x₂+⋯+x_n=r,
where xi ＞ ni for 1 ≤ i ≤ n. The number of positive integer solutions is___________.

7. (5%) Is the following a tautology:
Why?

8. (10%) How many of the following five statements are correct? ___________
ㆍLet V be a vector space. Let S be a subset of V. Let U be a subspace of V. IfS U,

then span(S) U.

ㆍIf R is a linearly dependent subset of a vector space, then x e span(R | [x])) holds

for each vector x R.

ㆍBased on any consistent axiom set for set theory, any vector space admits a basis.

ㆍBased on the standard ZFC axioms for set theory, each inner-product space admits

an orthonormal basis.

ㆍFor any complex vector spaces V and W with dim(W) <∞ , if T is a linear surjec-

tion from V to W, then dim(V)= nullity(T) + rank(T).

9.(10%) How many of the following five statements are correct? __________
ㆍThe solution set for a system of homogeneous linear equations having an m x n rational coefficient matrix is a rational vector space.

ㆍAn m x n complex matrix A is invertible if and only if A* is invertible.
ㆍIf A is an m x n rational matrix, then rank(A) = rank(A^t).
ㆍIf A is an m x n invertible real matrix, then nullity(A) = nullity(

).
ㆍIf A is an n X n complex matrix, then

10. (10%) How many of the following five statements are correct? ________
ㆍLet A be an n x n rational matrix. If the rational n-tuple vector space Qⁿ" is the direct sum of the eigenspaces of A, then A can be diagonalized.
ㆍIf A is an n x n complex matrix with A*A = AA*, then the eigenspaces of A* equal the eigenspaces of A. If A is an n x n real matrix with A^t = A, then the characteristic polynomial of A can be written as a product of degree-one polynomials with real coefficients.
ㆍ If A is an n x n complex matrix with At = A, then all eigenvalues of A are real.
ㆍIf A and B are unitarily equivalent n x n complex matrices, then the trace of A*A equals the trace of B*B.

11. (10%) How many zero entries are there in the inverse of the following matrix?__________

12. (10%) Give a basis for the vector space of the linear transformations from the vector space R³ of real triples to the vector space R? of real pairs:_____________.

108年 - 108 國立臺灣大學_碩士班招生考試_資訊工程學研究所:數學(含線性代數和離散數學)#106050

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