线性代数期中考试试题.doc

20102011学年第一学期线性代数期中考试试卷 2010.11Part 1Multiple-choice test ( 3 points/each)1. Let A= and be the cofactor of the (i, j) entry. Then _A .2 B. 1 C. 0 D. -22. If all the solutions of the system of equations are solutions of , then rank(A)_rank(B)A. = B. C .not deteremined D. 3. A sufficient and necessary condition under which the homogeneous linear equations has nonzero solutions is _A. rank(A)t B. st C. D. 5. Let , , which one of the following is right?_A. AB=BA B. C. D. none of the above is right6. Assume that A and B are square matrices with the same size, if , then_A. or B. and C. or D. none of the above is right 7.Determine which one of the following sets form subspaces of ?_A. B. C. D. 8. Let A be an matrix and . Then =_A B. .C. D. not determinedPart 2 Sutmnmy completion ( 3 points/each)1. Let and . If is a linear dependent set, then a= _. 2. For any matrix , let be an matrix, when equals _, we have .3. Given the vectors, the dimension of Span is _.4. If is a basis for . Then satisfies _.5. Let be a orthogonal matrix and and . Give all solutions for the linear system of in vector form. _.6. If , then equals _.7. The rank of the matrix A= equals_.8. The coordinator of a vector with respect to the basis in is _.Part 3. CALCULATE (5 points/each)1. Discuss the following system and give all solutions in vector form whenever the system has infinite many solutions.2. Find , if .3. Let ,. Find 4. Let two subspaces and Determined such that U=V and 5. Determine the nullspace of each of the following matrices.(a) (6) 6. Consider a nonhomogeneous system of linear equationsWhat value does take on such that the system has a solution?Part 4. PROVE1. Let .is a basis for the homogneeous linear system . If , then the set is a linear independent set. (7 points)2. (1) Let be an matrix, the elements of are real numbers. Prove: and have the same solutions. (4 points) (2). Prove that . (4 points)3. Suppose a system of fundamental solutions of the system is ,in other words, is a basis for nul