Linear Algebra Subspaces / Basis

Question

Let A = (a1 a2 a3 a4 a5), where ai is the i-th column of A, for i = 1,..., 5. It is known that

a1 = (1 -1 0 0), a3 = (-2 0 1 1), a5 = (-3 3 4 3).

Suppose A is row equivalent to R = (1 2 -1 2 -1; 0 1 -1 1 -1; 0 0 0 1 1; 0 0 0 0 0).

(a) What is the rank of A? Explain how you derive your answer.

(b) Show that { (2 2 2 1), (11 1 2 0), (5 4 3 2) } is a basis for the column space of A.

(c) Which of the following vectors v1 = (1 4 -3 7 0), v2 = (1 6 -5 3 1), v3 = (2 3 -1 1 -3), v4 = (1 1 0 4 3).

belongs to the row space of A? Explain how you derive your answer.

(d) Find a basis for the nullspace of A. Which is the nullity of A?

Thanks

Answer 1

(a) To determine the rank of matrix A, we count the number of nonzero rows in its row echelon form, R. From R, we see there are three nonzero rows. Hence, the rank of A is 3.

(b) To show that { (2 2 2 1), (11 1 2 0), (5 4 3 2) } is a basis for the column space of A, we need to demonstrate two things: linear independence and span.

Linear independence: We form a matrix B whose columns are the given vectors. We then reduce B to its row echelon form. If the resulting matrix has the same number of linearly independent rows as the original set of vectors, they are linearly independent.

Span: We need to show that any vector in the column space of A can be represented as a linear combination of the given vectors. This means we need to demonstrate that any column in A can be formed by scaling and adding the given vectors.

(c) To determine which of the vectors {v1, v2, v3, v4} belongs to the row space of A, we need to check if each vector can be expressed as a linear combination of the rows of A. If a vector can be expressed in this manner, it belongs to the row space of A.

(d) To find a basis for the nullspace of A, we need to solve the equation Ax = 0, where x is a vector in the nullspace. The nullspace of A consists of all vectors x that satisfy this equation. After solving for x, we form a basis from the linearly independent solutions. The nullity of A is the dimension of the nullspace, which is the number of linearly independent vectors in the basis for the nullspace.