Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra

Which of the following operations is NOT allowed in a linear system of equations?

• Scaling a variable by a constant.

• Adding scaled variables together.

• Raising a variable to an exponent.

• Having a constant on the right side of the equation.

A system of linear equations can be compactly written in the form Ax = v. What does 'A' represent in this equation?

• The vector of unknown variables.

• The matrix of constant coefficients.

• The vector of constant terms on the right side.

• The inverse of the system.

Geometrically, what does solving the equation Ax = v mean?

• Finding the transformation A that maps x to v.

• Finding the vector v that results from transforming x by A.

• Finding the vector x that, when transformed by A, lands on v.

• Finding the determinant of the matrix A.

If A represents a linear transformation, what is the defining property of its inverse, A⁻¹?

• A⁻¹ * A results in a transformation that squishes space to a lower dimension.

• A⁻¹ * A results in the identity transformation, which does nothing.

• A⁻¹ * A results in a transformation that rotates vectors by 90 degrees.

• A⁻¹ * A results in a transformation that doubles the length of vectors.

What condition must a linear transformation satisfy for its inverse to exist?

• Its determinant must be zero.

• Its determinant must be non-zero.

• It must squish space into a lower dimension.

• It must map multiple vectors to a single output.

What does the "rank" of a linear transformation represent?

• The number of input dimensions.

• The number of dimensions in the output of the transformation.

• The magnitude of the determinant.

• The number of basis vectors.

The "column space" of a matrix A refers to what?

• The set of all input vectors that map to the zero vector.

• The set of all possible output vectors resulting from the transformation A.

• The specific vectors that define the basis of the input space.

• The vectors that remain unchanged after the transformation.

What is the "null space" (or "kernel") of a linear transformation?

• The set of all vectors that are stretched by the transformation.

• The set of all vectors that are rotated by the transformation.

• The set of all vectors that map to the zero vector after the transformation.

• The set of all vectors that remain in their original position after the transformation.