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

Which of the following equations is not part of a linear system of equations?

• 2x + 5y + 3z = -3

• 4x + 0y + 8z = 0

• x^2 + y = 5

• 1x + 3y + 0z = 2

A system of linear equations can be represented 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 of the equations

• The solution vector

When a linear transformation A has a non-zero determinant, how many unique vectors 'x' will satisfy the equation Ax = v for a given vector 'v'?

• Zero

• One

• Infinitely many

• Two

What is the result of applying a linear transformation A, followed by its inverse transformation A⁻¹?

• A transformation that doubles the original vector

• A transformation that rotates the vector by 180 degrees

• The identity transformation, which leaves the vector unchanged

• A transformation that projects the vector onto an axis

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.