Question

select all the statements below that describe eigenvectors for the linear transformation . there may be more than one correct answer. a. b. c. d. any nonzero vector parallel to the -axis e. f. any nonzero vector parallel to the line g. any nonzero vector parallel to the line h. i. j. any nonzero vector parallel to the line the set of all eigenvectors of the linear transformation with eigenvalue together with the zero vector is the eigenspace . when described in words, the set is (select all that apply): a. the zero vector together with all vectors parallel to b. the zero vector together with all vectors parallel to the -axis c. the zero vector together with all vectors parallel to d. the zero vector together with all vectors parallel to e. the zero vector together with all vectors parallel to the -axis f. the zero vector together with all vectors parallel to the line the set of all eigenvectors of the linear transformation with eigenvalue together with the zero vector is the eigenspace . when described in set notation, the set is (select all that apply): a. b. c. d. note: you can earn partial credit on this problem. your score was score was successfully sent to the lms. you have attempted this problem 13 times. you received a score of 63% for this a

Answer 1

Eigenvectors of a linear transformation remain parallel to their original direction post-transformation, scaling by an eigenvalue, and form an eigenspace with the zero vector. A coordinate system projecting vectors onto the x-axis and y-axis is effective for two-dimensional vector problems which don't have parallel vectors.

Step-by-step explanation:

The concept in question is related to eigenvectors and eigenvalues of a linear transformation in a vector space, specifically concerning their geometric representation in two dimensions.

Eigenvectors are non-zero vectors that, after a linear transformation is applied, remain parallel to their original direction, only scaled by a corresponding eigenvalue. The set of all eigenvectors associated with a particular eigenvalue, combined with the zero vector, forms an eigenspace.

An eigenvector parallel to the x-axis would be a múltiple of the unit vector î, and any vector in the eigenspace would be a scalar multiple of this unit vector, including the null vector.

When multiple vectors in the problem are not parallel, it is efficient to use a coordinate system where vectors are projected onto the orthogonal x-axis and y-axis for analysis.

Given this information, to accurately select statements that describe eigenvectors for the linear transformation, one would need the specific linear transformation or its matrix representation.

However, the general principle is that any nonzero vector that, post-transformation, aligns with the original direction (scaled by the eigenvalue) is an eigenvector. For eigenspaces, including the zero vector, they consist of all scalar multiples of any eigenvector plus the zero vector itself.