Understanding the Null Space: Proving it as a Vector Space

A denotes an m×n matrix. Determine whether the statement is true or false. Justify your answer: A null space is a vector space.

The statement is true

The statement is true. The null space of a matrix is indeed a vector space.

To justify this, let’s first define what the null space is. The null space of a matrix A, denoted as null(A), is the set of all vectors x such that Ax = 0, where 0 is the zero vector. In other words, it is the set of all solutions to the equation Ax = 0.

To prove that the null space is a vector space, we need to show that it satisfies three properties: closure under addition, closure under scalar multiplication, and contains the zero vector.

1. Closure under addition: Let’s take two vectors x and y in the null space of A. This means that Ax = 0 and Ay = 0. Now, let’s consider the vector x + y. We need to show that (x + y) is also in the null space of A.

To do this, we can use the properties of matrix algebra:
A(x + y) = Ax + Ay (distributive property of matrix multiplication)
= 0 + 0 (since Ax = 0 and Ay = 0)
= 0

Therefore, (x + y) is also in the null space of A. This shows that the null space is closed under addition.

2. Closure under scalar multiplication: Let’s take a vector x in the null space of A, which means that Ax = 0. Now, consider the scalar k multiplied by x, represented as kx. We need to show that kx is also in the null space of A.

Again, using the properties of matrix algebra:
A(kx) = k(Ax) (associative property of scalar multiplication)
= k(0) (since Ax = 0)
= 0

Therefore, kx is also in the null space of A. This demonstrates that the null space is closed under scalar multiplication.

3. Contains the zero vector: The zero vector, denoted as 0, is always in the null space of any matrix A, since A0 = 0. Therefore, the null space contains the zero vector.

Since the null space of matrix A satisfies all three properties (closure under addition, closure under scalar multiplication, and contains the zero vector), it qualifies as a vector space. Hence, the statement that the null space is a vector space is true.

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