Lecture 6 Amanzhan.G,Erasyl.R

Which of the following statements defines a vector space?

• A collection of points with defined addition and multiplication

• A set of real numbers with addition and multiplication operations

• A set of objects with two operations satisfying specific axioms

Which of the following is NOT an operation defined on a vector space?

• Vector addition

• Scalar multiplication

• Vector division

• Zero vector addition

The zero vector is always a member of any subspace

• True

• False

• I dont no

If u is in the null space of matrix A, what happens when you multiply u by a scalar c?

• cu becomes the zero vector

• cu remains in the null space of A.

• cu is no longer a solution to

• The answer depends on the specific matrix A.

The column space of a matrix A represents

• All possible linear combinations of the rows of A

• All possible linear combinations of the columns of A

• The set of solutions to the equation Ax = 0.

• The number of rows and columns in the matrix A

The properties of additivity and homogeneity of a linear transformation T guarantee that

• T preserves vector addition and scalar multiplication

• T is a one-to-one function

• T is an onto function

If a set of vectors V is closed under addition and scalar multiplication, what can you conclude about V?

• V is a vector space

• V is a subspace of a larger vector space.

• V is not a set of vectors

The null space of a matrix A is the set of all vectors

• That are non-zero solutions to the equation Ax = 0

• That have a magnitude of zero.

• That are in the column space of A.

• That satisfy the equation Ax = 0

Which of the following statements is NOT always true about the column space of a matrix A?

• It is a subspace of R^m, where m is the number of rows in A

• It is closed under scalar multiplication

• It contains the zero vector

• It is closed under vector addition

The intersection of two subspaces H1 and H2 of a vector space V is always a subspace of V

• True

• False