* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download (pdf)
Capelli's identity wikipedia , lookup
Euclidean vector wikipedia , lookup
Quadratic form wikipedia , lookup
Tensor operator wikipedia , lookup
Determinant wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Eigenvalues and eigenvectors wikipedia , lookup
Matrix (mathematics) wikipedia , lookup
Singular-value decomposition wikipedia , lookup
System of linear equations wikipedia , lookup
Non-negative matrix factorization wikipedia , lookup
Covariance and contravariance of vectors wikipedia , lookup
Jordan normal form wikipedia , lookup
Orthogonal matrix wikipedia , lookup
Perron–Frobenius theorem wikipedia , lookup
Cartesian tensor wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Bra–ket notation wikipedia , lookup
Linear algebra wikipedia , lookup
Matrix multiplication wikipedia , lookup
Four-vector wikipedia , lookup
Math 314: Linear Algebra Spring 2006 Name Exam 1 1. The following matrix A is invertible. Find the 3 6 3 inverse. 2 0 6 0 4 1 Let b be the vector [2, 0, 3]. How many solutions does it have the system Ax = b? List all the solutions. What is the span of the columns of A? 2. Let A be the following matrix: 0 3 0 −3 9 2 2 0 2 8 . 0 2 2 0 10 and let R be the following matrix 1 0 0 2 1 0 1 0 −1 3 . 0 0 1 1 2 R is the reduced echelon form of A. Mark as true or false the following statements: • row(A) = row(R). • col(A) = col(R). • null(A) = null(R). Find a basis for the vector space col(A). List the free variables for the system Ax = b and find a basis for the vector space null(A). Find the rank(A). 3. Explain why the rows of a 3 × 5 matrix have to be linearly dependent. 4. Let A be a matrix wich is not the identity and assume that A2 = A. By contradiction show that A is not invertible. 5. Let u, v, w three linearly independent vectors. Are u − v, v − w, u − w linearly independent? Explain. 6. Let 3x + 5y + 7z = 0 be a plane. Is it the plane perpendicular to the line x = 4t + 1, y = −t + 9, z = −t + 190? 7. True or false: Given two matrices A and B is it true that (A + B)2 = A2 + 2AB + B 2 ? Good Luck. 1