If x = xo and x = x1 are both vector solution of Ax=b, then x1 - xo is in the Null Space of A.
True or False?
To determine if the statement is true or false, we need to understand some important concepts related to vector solutions of a matrix equation.
Let's start by defining the terms used in the statement:
- Ax = b: This represents a matrix equation, where A is a given matrix, x is a vector of unknowns, and b is a vector of constants.
- Vector solution: A vector x is considered a solution to the matrix equation Ax = b if, when the values of x are substituted into the equation, it satisfies the equation (A*x = b).
Next, recall the definition of the null space (or kernel) of a matrix:
- Null Space: The null space of a matrix A, denoted by Null(A), consists of all vectors x such that Ax = 0 (the zero vector).
Now, let's analyze the statement:
"If x = xo and x = x1 are both vector solutions of Ax = b, then x1 - xo is in the Null Space of A."
If xo and x1 are both vector solutions of Ax = b, it means that when we substitute xo and x1 into the equation, we get Axo = b and Ax1 = b. From this information alone, we cannot conclude that x1 - xo is in the Null Space of A.
To establish if x1 - xo is in the Null Space of A, we need to verify if A*(x1 - xo) = 0. Let's do this algebraically:
A*(x1 - xo) = Ax1 - Axo
Since x1 and xo are both vector solutions, we have:
Ax1 = b and Axo = b. Substituting these into the equation, we get:
A*(x1 - xo) = b - b
This simplifies to:
A*(x1 - xo) = 0
Therefore, we have shown that A*(x1 - xo) = 0, which means x1 - xo is in the Null Space of A.
Hence, the statement is true:
If x = xo and x = x1 are both vector solutions of Ax = b, then x1 - xo is in the Null Space of A.