The determinant | Chapter 6, Essence of linear algebra

A linear transformation is represented by the matrix [[3, 0], [0, 2]]. If a unit square (area 1) is transformed by this matrix, what will be the area of the resulting shape?

• 1

• 2

• 3

• 6

What does the determinant of a 2D linear transformation represent?

• The change in length of vectors.

• The factor by which areas are scaled.

• The angle of rotation of the space.

• The shift in the origin of the coordinate system.

What does a determinant of zero for a 2D linear transformation signify?

• The transformation rotates the space by 90 degrees.

• The transformation scales all areas by a factor of zero, effectively squishing all of space onto a line or a single point.

• The transformation doubles the area of all shapes.

• The transformation leaves all areas unchanged.

If a 2D linear transformation has a negative determinant, what does this indicate about the transformation?

• The transformation reduces the size of all shapes.

• The transformation increases the size of all shapes.

• The transformation inverts the orientation of space (flips it over).

• The transformation is not possible.

What happens to the determinant of a 2D linear transformation as the basis vectors become linearly dependent and then cross over each other?

• It remains constant at 1.

• It approaches zero and then becomes negative.

• It approaches infinity.

• It becomes undefined.

In three dimensions, what does the determinant of a linear transformation represent?

• The scaling factor of lengths.

• The scaling factor of areas.

• The scaling factor of volumes.

• The angle of rotation.

If the determinant of a 3D linear transformation is zero, what does this imply about the transformed space?

• The space is stretched uniformly in all directions.

• The space is rotated by 90 degrees.

• The space is squished onto a lower-dimensional object (like a plane, line, or point).

• The transformation is not possible.

What is the formula for the determinant of a 2x2 matrix with entries [[a, b], [c, d]]?

• ac - bd

• ad + bc

• ab - cd

• ad - bc