4.1 Find the eigenvalues and the corresponding eigenvectors of the matrix A.

1. Use Laplace Transform to solve the differential equation y"+4y'+3y=e'；y(0)=0,y'(0)=2.

2.The points A(1,-21), B(0,1,6) and C(-3,4,-2) form the vertices of a triangle. Calculate the angle between the line AB and the line from A to the midpoint P of . You do not need to give the explicit value of the angle. Just express the angle using inverse trigonometric functions. Hint: use dot product for two vectors AB and AP which is a scalar equal to the multiplication of the lengths of the two vectors and the cosine of the angle between the two vectors.

3. Find the plane containing the points A(1,21), B(-1,13) and C(-2,-2,-2). Hint: the cross product of two vectors is another vector that is orthogonal to the two vectors. 

4.2. Find matrix P that makes matrix A diagonal, that is, D= lAP where D is a diagonal matrix.

4.3 Find the n-th power of matrix A, that is, A". It is not necessary to explicitly compute . You can just express the result in terms of P and .

5. Let matrix that is composed of tirere column vestors Find the QR-decompostion of A, ie, A=OR where Q is a 3xs matix with ortionormal colann vectors and R is a 3x3 invertible upper triangular matrix. Hint: use Gram-Schmidt process to transform the basis vectors of the three column vectors in A into an orthogonal basis, and then normalize the orthogonal basis vectors to obtain an orthonormal basis corresponding to the column vectors q1,q2,q3 in matrix Q. The entry of matrix R is the inner product of vectors .

6. Let function f(x) be defined for -L sx sL. The Fourier series of f(x) isFind the coefficients of the Fourier series for the function f(x)= x -π ≤ x ≤ π .

6.2 Use the result of 6.1 to solve 

6. 6.1 Find the LU-factorization of the matrix

5. Find the inverse Laplace transform of

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