課程名稱︰線性代數
課程性質︰電機系大一必修
課程教師︰蘇柏青
開課學院：電資學院
開課系所︰電機系
考試日期（年月日）︰105/4/20
考試時限（分鐘）：100分鐘
試題 :
1. Consider an m*n matrix A. Ax=b is a system of linear equations, where x=[x1,x2,...,xn]^T represents n variables and b=[b1,b2,...,bm]^T represents m constants.
Answer whether the following statements are true or false. Justify your answer.
(a)(5%) If Ax=b is consistent for all b∈R^m, then n>=m.
(b)(5%) If there is at most one solution for Ax=b for all b∈R^m , then rank(A)=m.
(c)(5%) The set of all solutions of Ax=b is a subspace of R^n.
2. Let matrices A and B be the matrix representations of a linear operator T:R^n→R^n with respect to bases α={α1,α2,...,αn} and β={β1,β2,...,βn} respectively.
(a)(10%) Prove that the dimensions of the null spaces of A and B are identical, i.e., dim Null(A)=dim Null(B).
(b)(5%) Prove that det(A)=det(B).
3. Answer whether the following statements are true or false. Justify your answer.
(a)(5%) The mapping Φ:R^n→R^n defined by Φ(x)=[x]_α is a linear transformation, where [x]_α is the coordinate vector of x relative to a basis α of R^n.
(b)(5%) If T:R^n→R^n is onto then T is one-to-one.
4. (10%) Let V and W be subspaces of R^n. Let V+W={x+y:x∈V and y∈W}. Prove that V+W is a subspace of R^n.
5. (15%) Let A and B be both invertible matrices of size n*n. Answer whether the following statements are true of false. Justify your answer.
(a)(5%) A+B is also invertible.
(b)(5%) AB is also invertible.
(c)(5%) det(B)·(A^-1) is also invertible.
6. (15%) Consider the 2*4 matrix A=[1 3 5 7].
2 4 6 8
(a)(5%) Let R be the reduced row echelon form of A. Find R.
(b)(5%) Find an invertible matrix P such that PA=R.
(c)(5%) Let V be the solution set of Ax=0. Show that V is a subspace, and find a basis for V.
7. (20%)
(a)(5%) Let {v0,v1,v2} be a subset of R^2. Suppose {v1-v0,v2-v0} is linearly independent. Show that {v0-v1,v2-v1} is also linearly independent.
(b)(5%) With the same assumptions in (a), show that {v0-v2,v1-v2} is linearly independent.
(c)(10%) Let {v0,v1,...,vn} be a set of n+1 elements in R^n. Suppose {v1-v0,v2-v0,...,vn-v0} is linearly independent. Show that for any i∈{1,2,...,n}, the set {v0-vi,v1-vi,...,v(i-1)-vi}∪{v(i+1)-vi,...,vn-vi} is also linearly independent.
(Note: for notational consistency, a set with a beginning index larger than its ending index indicates an empty set, and a set with a beginning index equal to its ending index indicates a single-element set, e.g., {v(n+1)-vn,...,vn-vn}=Φ, and {v0-v1,v1-v1,...v(1-1)-v1}={v0-v1}.)