Linear Algebra (8110): Analyzing True/False Statements - Assignment 1

This document provides solutions to a Linear Algebra assignment that involves determining the truth or falsehood of various statements related to core concepts. The assignment covers topics such as eigenvectors and eigenvalues, properties of matrices, determinants, and diagonalizability. Each question requires a justification for the chosen answer, demonstrating an understanding of the underlying principles of linear algebra. The solutions reference standard textbooks on linear algebra, indicating a comprehensive approach to solving the problems. Desklib is a platform where students can find similar solved assignments and study resources to aid their learning.

Running Head: Linear Algebra 1
Linear Algebra
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Question 1 Solution
a) FALSE. v must have a nontrivial (non-zero) solution and |A−λ I|=0 .If that’s the case, v
is the eigenvector of A corresponding to the eigenvalueλ since ( A−λ I ) v=0.
b) FALSE. The converse is true. If there are n distinct eigenvalues (scalars) λ1 , λ2 … λn ,then
v1 , v2 … vnare linearly independent eigenvectors corresponding to the eigenvalues.
c) FALSE. Only for triangular matrices, the eigenvalues are the entries on the main
diagonal.
d) TRUE. The Eigen space of A is the null space of ( A−λ I )
Question 2 Solution
a) FALSE. The volume of the parallelepiped is the absolute value of the determinant.
Applying linear transformation from the unit cube, the volumes are multiplied by the
determinant.
b) FALSE. det AT =det A .
c) TRUE. From the definition, the multiplicity of a root r of a characteristic equation is the
algebraic multiplicity of r as an eigenvalue of A.
d) FALSE. Exchanging the columns of A will change the row operations on A.
Consequently, the characteristic equation will be different thus changing the eigenvalues
of A.
Question 3 Solution
a) FALSE. The eigenvectors must be linearly independent. For A to be diagonalizable, it
must strictly have n linearly independent eigenvectors.
b) FALSE. The eigenvalues can be repeated provided that the multiplicity of each
eigenvalue is equal to the basis of its eigenspace. However, if A has n distinct
eigenvalues, then A is diagonalizable because in that case, there will be n linearly
independent eigenvectors corresponding to the n distinct eigenvalues.
c) TRUE. The columns of P must be linearly independent eigenvectors of A if D is a
diagonal matrix. These eigenvectors correspond to the eigenvalues that are the diagonal
entries of matrix D.
d) FALSE. Diagonalization and being invertible are not directly related since having zero
eigenvector does not affect diagonalization. However, | A−λ I|=0 ,that is ( A−λ I )must be
invertible and singular.
Question 4 Solution

Question 5 Solution

Question 6 Solution

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Question 7 Solution

Question 8 Solution

References
Seymour, L. (2011). Schaum's Outline of Linear Algebra (4th ed.). New York: McGraw Hill.
Hohn, F. (2013). Elementary Matrix Algebra (1st ed.). New York: Dover Publications.
Ravinda, B, (2015), Linear Algebra and Linear Models (2nd ed). New York: Springer Publishing.