Linear Transformations and Their Properties

What is the null space of a matrix A?

• The set of all vectors X such that A * X equals zero.

• The set of all vectors B such that A * X equals B.

• The set of all vectors that are images of at least one vector under multiplication by A.

• The set of all vectors that are not transformed by A.

How is the kernel of a linear transformation defined?

• The set of all vectors that are transformed into the identity vector.

• The set of vectors in the co-domain that are images of at least one vector in the domain.

• The set of vectors in the domain that the transformation maps into the zero vector of the co-domain.

• The set of all vectors that are not transformed by the linear transformation.

What is the range of a linear transformation?

• The set of vectors in the domain that are mapped into the zero vector.

• The set of vectors in the co-domain that are images of at least one vector in the domain.

• The set of all vectors that are not transformed by the linear transformation.

• The set of all vectors that are transformed into the identity vector.

In the zero transformation, what is the kernel?

• The entire co-domain.

• The set of all non-zero vectors.

• The entire domain.

• Only the zero vector.

What is the range of the zero transformation?

• The set of all non-zero vectors.

• Only the zero vector.

• The entire co-domain.

• The entire domain.

For a projection onto the XY plane, what does the kernel consist of?

• All points on the Z axis.

• Only the zero vector.

• All points on the XY plane.

• All points in three-dimensional space.

What is the range of a rotation about the origin in the XY plane?

• The entire XY plane.

• Only the zero vector.

• The set of all non-zero vectors.

• The Z axis.

What must be proven to show that the kernel is a subspace of the domain?

• That the kernel is equal to the range.

• That the kernel contains all vectors in the domain.

• That the kernel is closed under addition and scalar multiplication.

• That the kernel is empty.

What is necessary to prove that the range is a subspace of the co-domain?

• That the range is empty.

• That the range is closed under addition and scalar multiplication.

• That the range contains all vectors in the co-domain.

• That the range is equal to the kernel.

What property of linear transformations is used to prove closure under addition for the kernel?

• Identity property.

• Inverse property.

• Homogeneity property.

• Additivity property.