Question

For any eigenvector X = t [x y z], what values can t take, where t ? ?

a) t can be any real number except zero
b) t can only be zero
c) t can be any positive real number
d) t can be any non-zero real number

Answer 1

The scalar t in the eigenvector X = t [x y z] can be any non-zero real number, as eigenvectors are non-zero vectors that are scaled by a linear transformation.

Step-by-step explanation:

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it. The definition of an eigenvector includes that it cannot be the null vector, which by definition has no direction and zero magnitude. For any eigenvector X = t [x y z], where x, y, and z are the components of the vector and t is a scalar, the scalar t can take any real value except zero. If t were zero, the vector would become the null vector, which is not an eigenvector as it violates the definition.

Therefore, the correct answer to the question is d) t can be any non-zero real number, as scaling by any non-zero real number would indeed produce an eigenvector of the original linear transformation.