Exercises and solutions

Author: bvanderlei

Linear_Transformations.ipynb

@@ -276,7 +276,7 @@
    "source": [
     "### Invertibility\n",
     "\n",
-    "When applying a given linear transformations, we often consider if it is possible to *reverse the transformation*.  That is, we would like to know if it is possible to map all the vectors in the output space back to vectors in the input space such that images get sent back to their preimages.  If this is possible we say that the transformation is invertible.  More formally, a linear transformation $L:\\mathbb{V}\\to\\mathbb{W}$ is said to be **invertible** if there exists another transformation $L^{-1}:\\mathbb{W}\\to\\mathbb{V}$, known as the inverse, such that $(L^{-1}\\circ L)(X) = X$ for any vector $X$ in $\\mathbb{V}$.  \n",
+    "When applying a given linear transformation, we often consider if it is possible to *reverse the transformation*.  That is, we would like to know if it is possible to map all the vectors in the output space back to vectors in the input space such that images get sent back to their preimages.  If this is possible we say that the transformation is invertible.  More formally, a linear transformation $L:\\mathbb{V}\\to\\mathbb{W}$ is said to be **invertible** if there exists another transformation $L^{-1}:\\mathbb{W}\\to\\mathbb{V}$, known as the inverse, such that $(L^{-1}\\circ L)(X) = X$ for any vector $X$ in $\\mathbb{V}$.  \n",
     "\n",
     "The transformation $L$ in **Example 2** is invertible since the matrix $A$ is invertible.  If $X$ is any vector in $\\mathbb{R}^3$ and $Y=AX$ is the image vector of $X$, it is possible to find $X$ given $Y$ by using $X=A^{-1}Y$.  "
    ]
@@ -424,6 +424,125 @@
    "source": [
     "## Code solution here."
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 5:** Create a Python function which implements the transformation: $S:\\mathbb{R}^3\\to\\mathbb{R}^3$, given below. Use the function to provide evidence whether the transformation is **linear** or not.\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "S \\left(\\left[\\begin{array}{r} v_1 \\\\ v_2 \\\\ v_3 \\end{array} \\right]\\right) = \n",
+    "\\left[\\begin{array}{c} v_1 + v_2 \\\\  1 \\\\ v_3+v_1 \\end{array} \\right]\n",
+    "\\end{equation}\n",
+    "$$\n",
+    "\n",
+    "Repeat for the transformation $T:\\mathbb{R}^3\\to\\mathbb{R}^3$ is now defined by\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "T \\left(\\left[\\begin{array}{r} v_1 \\\\ v_2 \\\\ v_3 \\end{array} \\right]\\right) = \n",
+    "\\left[\\begin{array}{c} v_1 + v_2 \\\\  0 \\\\ v_3+v_1 \\end{array} \\right].\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 6:** $L:\\mathbb{R}^3\\to\\mathbb{R}^2$ is a **Linear Transformation** . Find $L(kU+V)$ given that $k=7$, \n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "L(U) = \\left[\\begin{array}{r} 1 \\\\ 1  \\end{array} \\right]\\hspace{1cm}\n",
+    "L(V) = \\left[\\begin{array}{r} 3 \\\\ 1  \\end{array} \\right]\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 7:** Let $T:\\mathbb{R}^3 \\to \\mathbb{R}^2$ be defined by $T(X)= AX$, where \n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "A = \\left[\\begin{array}{rrr} 1 & 0 & 2 \\\\ 2 & 1 & 1  \\end{array}\\right].\n",
+    "\\end{equation}\n",
+    "$$\n",
+    "\n",
+    "Find all vectors $X$ that satisfy $T(X) = \\left[\\begin{array}{r} 1 \\\\ 2  \\end{array} \\right]\n",
+    "$. \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 8:** Let $M:\\mathbb{P}_1 \\to \\mathbb{P}_3$ be a transformation defined by $M(p(x)) = x^3 + p(x)$. Determine whether $M$ is linear or not. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 9:** Let $S:\\mathbb{P}_2 \\to \\mathbb{P}_3$ and $T:\\mathbb{P}_3 \\to \\mathbb{P}_5$ be two **linear transformations** defined by the rules given below. Define the composition $T\\circ S$ and determine whether it is linear or not. Explain why $S\\circ T$ is not defined.\n",
+    "\n",
+    "$S(p(x)) = x(p(x))$ \n",
+    "\n",
+    "$T(q(x)) = x^2(q(x))$ \n",
+    "\n",
+    "where $p(x)$ is a polynomial in $\\mathbb{P}_2$ and $q(x)$ is a polynomial in $\\mathbb{P}_3$. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
   }
  ],
  "metadata": {