To show that for any scalar c, the vector cw also satisfies the equation Ax = 0, we need to demonstrate that multiplying the matrix A by the vector cw results in a zero vector.
Given that Ax = 0, we know that multiplying the matrix A by the vector x yields a zero vector. We can express this as:
A * x = 0
Now, let's consider the vector cw, where c is a scalar. Multiplying the matrix A by cw gives:
A * (cw)
Using the properties of matrix multiplication and scalar multiplication, we have:
A * (cw) = (Ac) * w
Since Ac is the result of multiplying the matrix A by the scalar c, it is still a matrix. Therefore, we can rewrite the expression as:
(Ac) * w
Now, if we substitute cw with its equivalent expression (Ac) * w, we have:
(Ac) * w = (A * c) * w
Again, using the properties of matrix multiplication and scalar multiplication, we can write:
(A * c) * w = c * (A * w)
Since Ax = 0, we know that A * x = 0. Therefore, substituting x with w, we have:
c * (A * w) = c * 0
And finally:
c * 0 = 0
This shows that multiplying the matrix A by the vector cw results in a zero vector. Hence, for any scalar c, the vector cw also satisfies the equation Ax = 0.