推 itsweb:OP.. 05/03 12:34
課程名稱︰工程數學-線性代數
課程性質︰系定必修
課程教師︰馮蟻剛
開課系所︰電機系
考試時間︰2006.3.15
試題 :
1. Judge if the following statement are true or false. Give a concise proof to each true statement, and a counterexample to each false statement. (25%)
(a)If Ax=b is consistent, then the nullity of the augmented matrix [A b] equals the number of free variables in the general solution to Ax=b.
(b)If the set S is linearly independent, then no vector in S is a linear combination of the others.
(c)The determinant of any square matrix equals th product of the diagonal entries of its reduced row echelon form.
(d)The null space of any matrix equals the null space of its reduced row echelon form.
(e)For any m×n matrix A and n×m matrix B, det(AB)=det(BA).
2. For any tow subspace V and U of R^n, a set W={w| w=v+u, where v in V and u in U} is defined and denoted as V+U. In other words, V+Uis defined as the set of all vectors that are obtained by adding a vector in V and a vector in U.
(a)Prove that in general V+U is a subspace of R^n. (10%)
(b)suppose V∩U={0}, Bv is a basis of V, and Bu is a basis of U. Prove that the sat Bv∪Bu is a basis of V+U.(15%)
3. Let R be the reduced row echelon form of an arbitary m×n matrix A.
(a)Find the reduce row echelon of R^T and show that rank R^T = rank A. (15%)
(b)Use (a) to show that rank A^T = rank A.
4. Let T be the linear transformation of the orthogonal projection in R^3 on to the plane P. More specifically, for all v in R^3, T(v)=w asin the following figure, where w is the perpendicular projection imagge of v on the plane P.
(a)For P={x1-2x2+3x3=0}, select a basis S with two vector on P and a vector not on P, and compute [T]s. (15%)
(b)Prove that for any basis B of R^3 the value of det([T]B) is a constant and find the value. (10%)
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