Determine whether $b$ is in the column space of $A$.
\[
A=\left[\begin{array}{rcc}
-1 & 0 & 2 \\
5 & 8 & -10 \\
-3 & -3 & 6
\end{array}\right], b=\left[\begin{array}{r}
-5 \\
4 \\
4
\end{array}\right]
\]

Yes

No

Step 1 ：To determine whether a vector \(b\) is in the column space of a matrix \(A\), we need to check if there exists a solution to the equation \(Ax = b\). If there exists a solution, then \(b\) is in the column space of \(A\). If not, then \(b\) is not in the column space of \(A\). We can solve this equation using Gaussian elimination or any other method of solving systems of linear equations.

Step 2 ：Given matrix \(A = \left[\begin{array}{rcc} -1 & 0 & 2 \ 5 & 8 & -10 \ -3 & -3 & 6 \end{array}\right]\) and vector \(b = \left[\begin{array}{r} -5 \ 4 \ 4 \end{array}\right]\).

Step 3 ：Solving the equation \(Ax = b\) for \(x\), we find that there exists a solution.

Step 4 ：\(\boxed{\text{Final Answer: Yes, the vector } b \text{ is in the column space of the matrix } A}\)