Linear Transformations and Their Properties

What is the relationship between the kernel of a linear transformation and the null space of a matrix?

• They are unrelated concepts.

• The null space is a subset of the kernel.

• The kernel is a generalization of the null space.

• The kernel is always larger than the null space.

How is the rank of a linear transformation defined?

• As the sum of the dimensions of the domain and codomain.

• As the dimension of the range.

• As the dimension of the kernel.

• As the number of rows in the transformation matrix.

What does the dimension theorem for linear transformations state?

• Rank minus nullity equals the dimension of the codomain.

• Rank times nullity equals the number of columns.

• Rank plus nullity equals the dimension of the domain.

• Rank equals nullity.

In the example of the dilation operator, what is the rank of the transformation?

• Zero

• Four

• Three

• Two

What is the nullity of the dilation operator on 2x2 matrices?

• Zero

• Two

• One

• Four

In the evaluation transformation example, what is the dimension of the domain?

• Four

• Three

• Two

• One

What is the range of the evaluation transformation on polynomials?

• R to the fifth

• R to the fourth

• R cubed

• R squared

Why is the nullity of the evaluation transformation zero?

• Because the kernel contains only the zero vector.

• Because the codomain is R squared.

• Because the range is infinite.

• Because the domain is empty.

How many roots would a polynomial in the kernel of the evaluation transformation have?

• None

• Three

• Two

• One

What ensures that the range of the evaluation transformation is R cubed?

• The nullity being greater than the rank.

• The dimension theorem and the rank being equal to the dimension of the domain.

• The codomain being R squared.

• The kernel being non-zero.