SOLUTION: Suppose A is a 3 × 3 matrix and consider the set
{I3,A,A2,A3,...,A9}. Explain why this set must be linearly dependent.
Algebra ->
College
-> Linear Algebra
-> SOLUTION: Suppose A is a 3 × 3 matrix and consider the set
{I3,A,A2,A3,...,A9}. Explain why this set must be linearly dependent.
Log On
Question 1166866: Suppose A is a 3 × 3 matrix and consider the set
{I3,A,A2,A3,...,A9}. Explain why this set must be linearly dependent. Answer by ikleyn(52772) (Show Source):
The linear space of 3x3 matrices is 9-dimensional space over the base field.
The list in the problem represents 10 (ten) elements of this linear space.
If in any finite-dimensional (n-dimensional) space you have the set of elements ("vectors") with the number of elements
more than n, the dimension of the space, then this set of elements is linearly dependent over the base field
(which is a GENERAL fact from linear algebra).
