Matrix multiplication as composition | Chapter 4, Essence of linear algebra

Which of the following properties describes a linear transformation?

• Grid lines remain parallel and evenly spaced, and the origin remains fixed.

• Grid lines can become curved, but the origin remains fixed.

• Grid lines remain parallel, but spacing can vary, and the origin can shift.

• Grid lines can rotate, and the origin can shift.

How are the columns of a matrix representing a linear transformation determined?

• They are the original coordinates of the basis vectors.

• They are the transformed coordinates of the basis vectors.

• They are arbitrary values chosen to simplify calculations.

• They represent the scaling factors applied to the x and y axes.

What does multiplying a matrix by a vector achieve in the context of linear transformations?

• It calculates the determinant of the transformation.

• It applies the linear transformation represented by the matrix to the vector.

• It reverses the linear transformation.

• It scales the matrix by the vector's magnitude.

When two matrices are multiplied, what does the resulting matrix represent geometrically?

• The sum of the two individual transformations.

• The inverse of one of the transformations.

• The composition of the two linear transformations, applied from right to left.

• A single transformation that scales the original space.

Given matrices M2 = [[0, 2], [1, 0]] and M1 = [[1, -2], [1, 0]], what is the first column of the product M2 * M1?

• [0, 1]

• [2, 1]

• [1, 2]

• [1, 0]

Given matrices M2 = [[0, 2], [1, 0]] and M1 = [[1, -2], [1, 0]], what is the second column of the product M2 * M1?

• [0, -2]

• [-2, 0]

• [2, 0]

• [0, 2]

If M2 = [[a, b], [c, d]] and M1 = [[e, f], [g, h]], what is the resulting matrix M2 * M1?

• [[ae + bg, af + bh], [ce + dg, cf + dh]]

• [[ae + cf, be + df], [ag + ch, bg + dh]]

• [[ae + bg, ce + dg], [af + bh, cf + dh]]

• [[ae + cf, ag + ch], [be + df, bg + dh]]

Which of the following statements about matrix multiplication is true?

• Matrix multiplication is always commutative (A * B = B * A).

• Matrix multiplication is always associative ( (A * B) * C = A * (B * C) ).

• Matrix multiplication is never associative.

• Matrix multiplication is always commutative and associative.