{"action":"created","comment":{"url":"https://api.github.com/repos/siefkenj/IBLLinearAlgebra/pulls/comments/199351746","pull_request_review_id":133441314,"id":199351746,"node_id":"MDI0OlB1bGxSZXF1ZXN0UmV2aWV3Q29tbWVudDE5OTM1MTc0Ng==","diff_hunk":"@@ -2462,25 +2462,66 @@ \\section*{Matrices}\n \t\t\\mat{1&-3&0\\\\0&0&1\\\\0&0&0}\\mat{x\\\\y\\\\z}=\\vec b.\n \t\\]\n \t\\begin{parts}\n-\t\t\\item If $\\vec b=\\mat{1\\\\2\\\\3}$, is the set of solutions to this system a \n-\t\tpoint, line, plane, or other?\n-\t\t\\item If $\\vec b=\\mat{1\\\\1\\\\0}$, is the set of solutions to this system a \n-\t\tpoint, line, plane, or other?\n+\t\t\\item If $\\vec b=\\mat{1\\\\2\\\\3}$, is the set of solutions to this system\n+\t\t\ta point, line, plane, or other?\n+\t\t\t\\begin{solution}\n+\t\t\t\tThis system has no solutions, since the third equation would be\n+\t\t\t\t$0=3$, which is impossible. \n+\t\t\t\\end{solution}\n+\t\t\\item If $\\vec b=\\mat{1\\\\1\\\\0}$, is the set of solutions to this system\n+\t\t\ta point, line, plane, or other?\n+\t\t\t\\begin{solution}\n+\t\t\t\tA line.\tThe system would be\n+\t\t\t\t\\begin{align*}\n+\t\t\t\t\tx - 3y &= 1 \\\\\n+\t\t\t\t\tz &= 1 \\\\\n+\t\t\t\t\t0 &= 0\n+\t\t\t\t\\end{align*}\n+\t\t\t\t$\\vec x=\\mat{x\\\\y\\\\z}$ satisfies this system if and only if it is\n+\t\t\t\ton the line $\\vec x = t \\mat{3\\\\1\\\\0} + \\mat{1\\\\0\\\\1}$.\n+\t\t\t\\end{solution}\n \t\\end{parts}\n \n \t\\question\n-\tLet $\\vec d_1=\\mat{1\\\\1\\\\2}$ and\n-\t$\\vec d_2=\\mat{-1\\\\1\\\\0}$. \n+\tLet $\\vec d_1=\\mat{1\\\\1\\\\2}$ and $\\vec d_2=\\mat{-1\\\\1\\\\0}$. \n \tLet $\\mathcal P$ be the plane given in vector form by $\\vec x=t\\vec d_1+s\\vec d_2$.\n \tFurther, suppose $M$ is a matrix so that $M\\vec r\\in\\mathcal P$ for any $\\vec r$.\n \t\\begin{parts}\n \t\t\\item How many rows does $M$ have?\n+\t\t\t\\begin{solution}\n+\t\t\t\tThree. It must have three rows in order for $M\\vec r$ to be an\n+\t\t\t\telement of $\\R^3$. \n+\t\t\t\\end{solution}\n \t\t\\item Find such an $M$.\n-\t\t\\item Find necessary and sufficient conditions (phrased as equations) for $\\vec n$\n-\t\t\tto be a normal vector for $\\vec p$.\n-\t\t\\item Find a matrix $K$ so that solutions\n-\t\t\tto $K\\vec x=\\vec 0$ are normal vectors for $\\mathcal P$.\n-\t\t\tHow do $K$ and $M$ relate?\n+\t\t\t\\begin{solution}\n+\t\t\t\t$M=\\mat{1&-1&0\\\\1&1&0\\\\2&0&0}$ is one possible answer, since 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