Eigenvectors and Eigenspaces Concepts

This video tutorial explains how to determine the eigenvectors corresponding to the eigenvalues of a 2x2 matrix. It covers the process of finding eigenvectors for two specific eigenvalues, -3 and 2, by solving the eigenvector equation. The tutorial also discusses the concept of eigenspace and provides examples to illustrate the calculations. The video aims to help viewers understand the relationship between eigenvalues, eigenvectors, and eigenspaces in linear algebra.

What is the first step in finding eigenvectors once the eigenvalues are known?

• Transpose the matrix

• Add the eigenvalue to the matrix

• Solve the eigenvector equation

• Multiply the matrix by the eigenvalue

For lambda = -3, what is the first row of the coefficient matrix?

• Four plus three times one

• Negative seven plus three times zero

• Negative five plus three times one

• Two plus three times zero

What does it mean if x2 is a free variable in the context of eigenvectors?

• x2 is equal to x1

• x2 must be zero

• x2 can be any value

• x2 is negative

What is a basic eigenvector for lambda = -3?

• Vector 2 2

• Vector 1 0

• Vector 1 1

• Vector 0 1

What is included in the eigenspace of a matrix corresponding to a given eigenvalue?

• Only zero vectors

• Only scalar multiples of the eigenvalue

• Only non-zero vectors

• All vectors including the zero vector

What is the first step in finding eigenvectors for lambda = 2?

• Multiply matrix A by lambda

• Subtract lambda times the identity matrix from matrix A

• Add lambda to the identity matrix

• Transpose matrix A

For lambda = 2, what is the form of the eigenvectors?

• s times the vector 1 1

• s times the vector 2 7

• s times the vector 0 1

• s times the vector 1 0

What is a basic eigenvector for lambda = 2?

• Vector 1 1

• Vector 2 7

• Vector 1 0

• Vector 0 1

What is the eigenspace of matrix A corresponding to lambda = 2?

• All scalar multiples of the vector 1 0

• All scalar multiples of the vector 0 1

• All scalar multiples of the vector 2 7

• All scalar multiples of the vector 1 1

What must be true for a vector to be considered an eigenvector?

• It must be orthogonal to the matrix

• It must be a unit vector

• It cannot be the zero vector

• It must be the zero vector