Exploring Linear Transformations and Matrices

What is a linear transformation in the context of linear algebra?

• A function that takes in a scalar and outputs a vector.

• A function that takes in a vector and outputs a scalar.

• A function that takes in a matrix and outputs a vector.

• A function that takes in a vector and outputs another vector.

Which of the following is NOT a property of a linear transformation?

• Lines can become curved.

• The origin remains fixed.

• Grid lines remain parallel and evenly spaced.

• All lines remain straight.

How can you describe a linear transformation numerically?

• By using the coordinates of the diagonal vectors.

• By recording where the two basis vectors land.

• By recording the coordinates of all vectors.

• By using only the coordinates of the origin.

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

• A new vector that is a linear combination of the origin and the basis vectors.

• A new vector that is a linear combination of the diagonal vectors.

• A new vector that is a linear combination of the transformed basis vectors.

• A new vector that is a linear combination of the original basis vectors.

What does a 2x2 matrix represent in the context of linear transformations?

• The coordinates of the origin.

• The coordinates of all vectors.

• The coordinates of the transformed basis vectors.

• The coordinates of the diagonal vectors.

What happens to the vector (1, 0) after a 90-degree counterclockwise rotation?

• (0, 1)

• (0, -1)

• (1, 1)

• (-1, 0)

In a shear transformation where i-hat remains fixed, where does j-hat move?

• (1, 0)

• (-1, 0)

• (0, 1)

• (1, 1)

What does it mean if the vectors that i-hat and j-hat land on are linearly dependent?

• The transformation is not linear.

• The transformation squishes all of 2D space onto a line.

• The transformation keeps all lines straight.

• The transformation moves the origin.

What is the main takeaway about matrices in the context of linear transformations?

• Matrices are just a collection of numbers.

• Matrices are used to describe the origin.

• Matrices represent transformations of space.

• Matrices are used to describe diagonal vectors.

What will be discussed in the next video?

• Multiplying two matrices together

• Determinants

• Eigenvalues

• Change of basis