Transformation Matrices and Vector Effects

The video tutorial explains how to match each 3x3 matrix with its effect on vectors in R3. It covers four matrices, describing their transformations as projections, rotations, and reflections. The first matrix projects vectors onto the line y=x in the xy-plane. The second matrix rotates vectors 180 degrees about the x-axis. The third matrix reflects vectors across the plane x=z, and the fourth matrix reflects vectors across the xz-plane. Graphical representations are used to illustrate these transformations.

What is the effect of a 3x3 matrix on vectors in R3?

• It transforms vectors from R2 to R3.

• It transforms vectors from R3 to R2.

• It transforms vectors from R3 to R3.

• It transforms vectors from R2 to R2.

Which vector is transformed into the zero vector in the first transformation matrix?

• e sub 4

• e sub 3

• e sub 2

• e sub 1

How is the first transformation matrix best described?

• A rotation about the x-axis

• A reflection across the plane x = z

• A reflection across the plane y = 0

• A projection onto the line y = x in the xy-plane

What happens to the vector e sub 1 in the second transformation matrix?

• It is rotated 90 degrees.

• It becomes zero.

• It remains unchanged.

• It is reflected across the y-axis.

What is the transformation effect on vectors e sub 2 and e sub 3 in the second matrix?

• They are rotated 90 degrees.

• They are reflected across the x-axis.

• They are unchanged.

• They are rotated 180 degrees about the x-axis.

How is the third transformation matrix best described?

• A reflection across the plane y = 0

• A projection onto the line y = x

• A reflection across the plane x = z

• A rotation about the y-axis

In the third transformation matrix, which vectors swap positions?

• e sub 1 and e sub 3

• e sub 1 and e sub 4

• e sub 2 and e sub 3

• e sub 1 and e sub 2

What is the transformation effect on the vector e sub 2 in the fourth matrix?

• It becomes zero.

• It is reflected across the xz-plane.

• It remains unchanged.

• It is rotated 180 degrees.

How is the fourth transformation matrix best described?

• A rotation about the z-axis

• A reflection across the plane x = z

• A projection onto the line y = x

• A reflection across the plane y = 0

Which vector remains unchanged in the fourth transformation matrix?

• e sub 4

• e sub 2

• e sub 3

• e sub 1