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8. (15%) Let x(t) be a length-3 vector of functions in t that satisfies the following system of linear differential equations:

Find the solution x(t).

相關申論題

(a) (3%) Suppose Ax = b is a consistent system and v1 and v2 are two solutions, where A is an m X n matrix. Then v = v1 - V2 must be a solution for the homogeneous equation Ax = 0.

(b) (3%) Let A and B be m x n matrices. If Ax = Bx for all x , then A and B are equal.

(c) (3%) Let A be an n X n matrix. If A is invertible, then any subset of formed by the columns vectors of A is linearly independent.

(d) (3%) Let S = be a subset of and T : be a linear transfor- mation. When is linear independent, S is not necessarily linearly independent.

(e) (3%) Let A be an m x n matrix. If the number of pivot numbers of A is less than m, then Ax = b has more than one solution.

2. (10%) Let W1, W2,... , be subspaces of such that is a subset of is the orthogonal complement of , for i = 1,2, ,m - 1. Suppose the orthogonal projection matrix for . Determine the orthogonal projection matrix for V =

3.(a) (8%) Find the eignevalues and eigenvectors for

(b) (3%) How can you tell whether a matrix is invertible from its eigenvalues?

(c) (3%) How can you tell whether two matrices are similar from their eigenvalues?

4. (a) (8%) For the following matrix, find the bases for its row space and nullspace.

相關試卷

101年 - 101 國立交通大學_碩士班考試入學試題_電機工程學系：微分方程與線性代數#105738

101年 · #105738

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