Vector Transformations and Linear Functions

The video explores how 2x2 matrices can define transformations on the coordinate plane, such as reflections, dilations, and rotations. It demonstrates using vectors and identity matrices to achieve these transformations, encouraging experimentation with matrix manipulation to understand linear transformations better.

What is the primary purpose of using a 2x2 matrix in transformations?

• To define transformations on the coordinate plane

• To solve linear equations

• To calculate determinants

• To find eigenvalues

In the identity matrix, what does the red vector (1,0) represent?

• A vector in the y direction

• A vector in the x direction

• A zero vector

• A diagonal vector

How does the blue vector change during a reflection about the x-axis?

• It remains unchanged

• It becomes (0,-1)

• It becomes (0,1)

• It becomes (1,0)

What is the effect of changing the second column of the identity matrix to (0,-1)?

• It results in a reflection over the y-axis

• It results in a dilation

• It results in a reflection over the x-axis

• It results in a 90-degree rotation

What happens to the vectors during a dilation transformation by a factor of 0.5?

• They rotate 90 degrees

• They become half as long

• They remain the same

• They double in length

What is the result of setting both vectors in the matrix to (0.5,0)?

• A translation

• A reflection

• A dilation

• A rotation

What is the new position of the red vector after a 90-degree clockwise rotation?

• (-1,0)

• (0,-1)

• (0,1)

• (1,0)

Which transformation keeps the origin fixed and maps lines to lines?

• Projective transformation

• Linear transformation

• Affine transformation

• Non-linear transformation

What happens to the blue vector during a 90-degree clockwise rotation?

• It becomes (1,0)

• It becomes (0,1)

• It becomes (-1,0)

• It becomes (0,-1)

What is a key characteristic of a linear transformation?

• It maps the origin to a new point

• It maps lines to curves

• It maps lines to lines

• It changes the dimensionality