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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Diﬀerential Equations Week 11 Worksheet (April 18, 2005): Eigenvalue and Eigenvector Spring 200405 Q1. (Maximal independent vectors) Show that the ﬁve vectors in R3 : v1 = (1, 2, 1), v2 = (−1, 1, 0), v3 = (3, −2, 4), v4 = (0, −2, −3), v5 = (2, −3, 4) are linearly dependent. Then ﬁnd maximal number of linearly independent vectors from the ﬁve. 1 Q2. (Basis of R3 ) Consider the matrix A = 1 0 (a) Nul A, (b) Col A, (c) Row A. 1 1 0 0 0 1 2 3. Find linearly independent vectors that span 1 Q3. (Basis of R3 ) Determine the vectors v1 = (1, −2, 1), v2 = (2, −1, 3), v3 = (−1, 3, −1) form a basis of the Euclidean space R3 . Q4. (Basis of R3 ) Find condition on a such that the vectors v1 = (1, 2, −1), v2 = (2, 3, a), v3 = (a, 3, 1) form a basis of R3 . Q5. (Basis of Rn ) We illustrate the following fact: a basis of Rn must contain exactly n vectors in Rn . More or less number of vectors is not permissible. (a) Show that m vectors in Rn must be linearly dependent if m > n. (b) Show that m vectors in Rn do not span Rn if m < n. Q6. (Eigenvalues and eigenvectors) Determine whether the vectors v1 = (1, 2), v2 = (3, 4), v3 = (5, 6) are eigenvectors of the given matrix A. If so, what are the eigenvalues? A= −2 4 3 . −1 Q7. (Diagonalization) Verify the vectors v1 = (1, −1, 1), v2 = (1, 1, 0), v3 = (1, 0, −1) form a basis of eigenvectors of the given matrix A. Then ﬁnd a diagonalization of the matrix. 0 1 −1 0 1 . A= 1 −1 1 0 Q8. (Diagonalizability) Show that if A is diagonalizable, then A2 is also diagonalizable. ...
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 Spring '10
 KCC
 Math, Linear Algebra, Algebra, Equations, Vectors

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