wksht11 - MATH 152 L1 & L2 Applied Linear Algebra...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Week 11 Worksheet (April 18, 2005): Eigenvalue and Eigenvector Spring 2004-05 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. ...
View Full Document

Page1 / 2

wksht11 - MATH 152 L1 &amp;amp; L2 Applied Linear Algebra...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online