Chapter 3.1, Problem 42E

Express the image of the matrix $A=\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 4\\ 1& 3& 5& 2\\ 1& 4& 7& 0\end{array}\right]$ as the kernel of a matrix B. Hint: The image of A consists of all vectors $\stackrel{\to }{y}$ in ${ℝ}^4$ such that the system $A\stackrel{\to }{x}=\stackrel{\to }{y}$ is consistent. Write this system more explicitly: $\left|\begin{array}{l}{x}_1+x_2+x_3+6x_4=y_1\\ x_1+2x_2+3x_3+4x_4=y_2\\ x_1+3x_2+5x_3+2x_4=y_3\\ x_1+4x_2+7x_3=y_4\end{array}\right|$ .

Now, reduce rows:

$\left|\begin{array}{l}{x}_1-x_3+8x_4=4y_3-3y_4\\ x_2+2x_3-2x_4=-y_3+y_4\\ 0=y_1-3y_3+2y_4\\ 0=y_2-2y_3+y_4\end{array}\right|$

For which vectors $\stackrel{\to }{y}$ is this system consistent? The answer allows you to express im(A) as the kernel of a $2\times 4$ matrix B.