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. A null space is indeed a vector space.
To prove this, we need to first understand what a null space is. The null space of a matrix A, denoted as N(A), is the set of all vectors x such that Ax = 0, where 0 is the zero vector.
Now, let’s examine the properties that need to be satisfied for a set to be considered a vector space.
1. Closure under addition: For any two vectors u and v in N(A), the sum u + v must also be in N(A). This can be shown as follows:
If u and v are in N(A), then Au = Av = 0.
Adding these equations, we get Au + Av = 0 + 0, which simplifies to A(u + v) = 0.
Therefore, u + v is also in N(A), satisfying closure under addition.
2. Closure under scalar multiplication: For any vector u in N(A) and any scalar c, the product c * u must also be in N(A). Again, we can prove this:
If u is in N(A), then Au = 0.
Multiplying both sides of this equation by c, we get Acu = 0.
Thus, c * u is also in N(A), satisfying closure under scalar multiplication.
3. Existence of the zero vector: The zero vector, denoted as 0, is always in N(A) since A0 = 0.
4. Existence of additive inverses: For any vector u in N(A), the additive inverse -u must also be in N(A). This can be shown as follows:
If u is in N(A) and Au = 0, then multiplying both sides by -1, we get -Au = -0.
Simplifying, we have A(-u) = 0.
Therefore, -u is also in N(A), satisfying the existence of additive inverses.
Since the null space satisfies all four properties required for a set to be considered a vector space, we can conclude that the null space is indeed a vector space.
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