Understanding Linear Transformations and Eigenvectors

The video tutorial explains the linear transformation of a unit circle using matrix A, which has eigenvalues 3 and 1 with corresponding eigenvectors. It describes how the transformation affects the unit circle, stretching it along the line y=x by a factor of 3 while leaving it unchanged along y=-x. The tutorial concludes with an analysis of graphs to identify the correct transformation.

What is the result of applying a linear transformation to a vector using a matrix?

• The vector is scaled and possibly rotated.

• The vector is translated.

• The vector remains unchanged.

• The vector is rotated.

Which of the following is true about eigenvectors?

• They are vectors that always have a magnitude of one.

• They are vectors that always lie on the x-axis.

• They are vectors that do not change direction under a transformation.

• They are always perpendicular to each other.

What is the eigenvalue associated with the vector (2, 2)?

• 1

• 2

• 0

• 3

What is the eigenvalue associated with the vector (-3, 3)?

• 2

• 0

• 1

• 3

What happens to the vector (2, 2) under the given transformation?

• It is scaled by a factor of 1.

• It remains unchanged.

• It is scaled by a factor of 3.

• It is rotated by 90 degrees.

How does the vector (-3, 3) transform under the given matrix?

• It is scaled by a factor of 3.

• It is scaled by a factor of 0.

• It remains unchanged.

• It is scaled by a factor of 2.

What is the effect of the transformation on the unit circle along the line y = x?

• It is stretched by a factor of 3.

• It is compressed by a factor of 3.

• It is rotated by 45 degrees.

• It remains unchanged.

Why does the unit circle not stretch along the line y = -x?

• Because the eigenvalue is 1.

• Because the eigenvalue is 0.

• Because the eigenvalue is 3.

• Because the eigenvalue is 2.

Which line represents the direction along which the unit circle is stretched?

• y = 2x

• y = -x

• y = 0

• y = x

What is the correct graphical representation of the transformed unit circle?

• Stretched along both y = x and y = -x

• Not stretched at all

• Stretched along y = x only

• Stretched along y = -x only