The equation Ax = b is homogeneous if the zero vector is a solution.Is the statement true or false?
The statement is true
The statement is true.
To understand this, let’s first define what a homogeneous equation is. A linear equation is said to be homogeneous if the right-hand side, which is the vector b in the equation Ax = b, is a zero vector (i.e., all its elements are zero). In other words, a linear equation is homogeneous if it can be written in the form Ax = 0.
Now, if the zero vector is a solution to the equation Ax = b, this means that when we substitute the zero vector for x, the equation holds true. In mathematical notation, this can be represented as A(0) = b, which simplifies to the zero vector being equal to b (0 = b).
Since the zero vector is a vector where all its elements are zero, this implies that the right-hand side of the equation (which is b) must also be a zero vector. Hence, the equation Ax = b is homogeneous in this case.
Therefore, the statement is true – the equation Ax = b is homogeneous if the zero vector is a solution.
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