Question

Let a be an m n matrix. use the steps below to show that a vector x in rn satisfies ax d 0 if and only if a tax d 0. this will show that nul a d nul a ta.

a. show that if ax d 0, then a tax d 0.
b. suppose a tax d 0. explain why x ta taxd 0, and use this to show that ax d

Answer 1

To show that x satisfies Ax = 0 if and only if A^TAx = 0, follow the given steps: multiply both sides of the equation by A^T, then multiply both sides by x^T, simplify the equation using matrix properties, conclude that Ax = 0, and establish the if and only if condition.

Step-by-step explanation:

To show that a vector x in R^n satisfies Ax = 0 if and only if A^TAx = 0, we can follow the given steps:

1. If Ax = 0, then multiply both sides of the equation by A^T to obtain A^TAx = 0.
2. Suppose A^TAx = 0. We have to show that Ax = 0. To do this, multiply both sides of the equation by x^T, resulting in x^TA^TAx = 0.
3. Using the properties of matrix multiplication, we can simplify x^TA^TAx = 0 to (Ax)^T(Ax) = 0.
4. Since (Ax)^T(Ax) = 0, it implies that the dot product of Ax with itself is zero. And the only vector whose dot product with itself is zero is the zero vector.
5. Therefore, Ax = 0, and we have shown that if Ax = 0, then A^TAx = 0.

By establishing both directions of the if and only if condition, we can conclude that nul(A) = nul(A^TA).