Exercises added

Author: bvanderlei

Matrix_Representations.ipynb

@@ -142,7 +142,7 @@
    "source": [
     "If we keep in mind that there can be at most one pivot per column and one pivot per row, we can make a further statement about these properties based on the relative size of $m$ and $n$.  If $T:\\mathbb{R}^n\\to\\mathbb{R}^m$, then $T$ cannot be onto $\\mathbb{R}^m$ if $m>n$, and cannot be one-to-one if $m<n$.  Only when $m=n$ is it *possible* for $T$ to be invertible.\n",
     "\n",
-    "Let's look at our examples.  In **Example 1**, the matrix $[T]$ is $4\\times 2$.  We know the matrix can have at most two pivots (at most one per column), so cannot have a pivot in each row.  To see if there is are indeed pivots in each column, we can compute the RREF."
+    "Let's look at our examples.  In **Example 1**, the matrix $[T]$ is $4\\times 2$.  We know the matrix can have at most two pivots (at most one per column), so it cannot have a pivot in each row.  To see if there are pivots in each column, we can compute the RREF."
    ]
   },
   {
@@ -154,10 +154,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[1 0]\n",
-      " [0 1]\n",
-      " [0 0]\n",
-      " [0 0]]\n"
+      "[[1. 0.]\n",
+      " [0. 1.]\n",
+      " [0. 0.]\n",
+      " [0. 0.]]\n"
      ]
     }
    ],
@@ -279,6 +279,87 @@
    "source": [
     "## Code solution here."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 3:** $L:\\mathbb{R}^3\\to\\mathbb{R}^2$ is a **Linear Transformation**. Find $L(X)$ given the following vectors.\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "L\\left(\\left[\\begin{array}{r} 1\\\\0\\\\0 \\end{array}\\right]\\right)= \\left[\\begin{array}{r} 2\\\\0 \\end{array}\\right] \\hspace{1cm}  \n",
+    "L\\left(\\left[\\begin{array}{r} 0\\\\1\\\\0 \\end{array}\\right]\\right)= \\left[\\begin{array}{r} 1\\\\3 \\end{array}\\right] \\hspace{1cm}  \n",
+    "L\\left(\\left[\\begin{array}{r} 0\\\\0\\\\1 \\end{array}\\right]\\right)= \\left[\\begin{array}{r} 1\\\\2 \\end{array}\\right] \\hspace{1cm}\n",
+    "X = \\left[\\begin{array}{r} 4\\\\5\\\\3 \\end{array}\\right]\n",
+    "\\end{equation}\n",
+    "$$\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 4:** The standard matrix representation of a **linear transformation** $S$ is given below. Determine the input and output space of $S$ by looking at the dimensions of $\\left[S\\right]$. Determine whether $S$ is an invertible transformation.\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "\\left[S\\right] =\\left[\\begin{array}{rr} 2 & 0 & 3 & 8\\\\ 0 & 1 & 9 & 4 \\end{array}\\right]  \n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 5:** The **linear transformation** $W:\\mathbb{R}^3\\to\\mathbb{R}^3$ is an invertible transformation. Find $X$.\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "\\left[W\\right] =\\left[\\begin{array}{rr} 1 & 1 & 0\\\\ 1 & 2 & 2 \\\\ 2 & 1 & 3 \\end{array}\\right] \\hspace{1cm}\n",
+    "W(X) = \\left[\\begin{array}{r} 3 \\\\ 11 \\\\ 13 \\end{array}\\right]\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 6:** Let $T:\\mathbb{R}^3\\to\\mathbb{R}^3$ be a **linear transformation**. Given that there are two vectors in $\\mathbb{R}^3$ which get mapped to the same vector in $\\mathbb{R}^3$, what can you say about the number of solutions for  $[T]X=B$?  Explain why $T$ is not an invertible transformation."
+   ]
   }
  ],
  "metadata": {