Exercises and solutions

Author: bvanderlei

Vector_Space_Examples.ipynb

@@ -159,15 +159,151 @@
     "\\end{eqnarray*}\n",
     "$$\n",
     "\n",
-    "These algebraic combinations of functions satisfy all the requirements to make $C\\left[0,1\\right]$ a vector space.  This space is quite different from the other examples in that it does not have a finite dimension.  Any basis for $C\\left[0,1\\right]$ must contain an infinite number of functions.  For this reason, we cannot easily do calculations such as those in the previous example to determine if a set of functions are linearly indpendent.  We will however make use of this example in a later application. \n"
+    "These algebraic combinations of functions satisfy all the requirements to make $C\\left[0,1\\right]$ a vector space.  This space is quite different from the other examples in that it does not have a finite dimension.  Any basis for $C\\left[0,1\\right]$ must contain an infinite number of functions.  For this reason, we cannot easily do calculations such as those in the previous example to determine if a set of functions are linearly independent.  We will however make use of this example in a later application. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exercises"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 1:** Determine whether or not the set of polynomials $\\{p_1, p_2, p_3\\}$ is a basis for $\\mathbb{P}_2$.\n",
+    "\n",
+    "$$\n",
+    "\\begin{eqnarray*}\n",
+    "p_1 & = & 3x^2 + 2x + 1 \\\\\n",
+    "p_2 & = & 2x^2 + 5x + 3 \\\\\n",
+    "p_3 & = & 6x^2 + 4x  +5 \n",
+    "\\end{eqnarray*}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 2:** Find the coordinates of $p_4$ with respect to the basis $\\alpha\\ = \\{p_1, p_2, p_3\\}$. \n",
+    "\n",
+    "$$\n",
+    "\\begin{eqnarray*}\n",
+    "p_1 & = & x^2 + x + 2 \\\\\n",
+    "p_2 & = & 2x^2 + 4x + 0 \\\\\n",
+    "p_3 & = & 3x^2  + 2x      +1 \\\\\n",
+    "p_4 & = & 11x^2 + 13x + 4\n",
+    "\\end{eqnarray*}\n",
+    "$$"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [],
-   "source": []
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 3:** Demonstrate that a set of four polynomials in $\\mathbb{P}_4$ cannot span $\\mathbb{P}_4$ through a computation."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 4:** The set of matrices $\\{A, B\\}$ form a basis for a subspace of $\\mathbb{M}_{2\\times 2}$. Find a matrix which is in the subspace (but is not $A$ or $B$) and a matrix which is not in the subspace. Verify your answer.\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "A = \\left[ \\begin{array}{ccc} 1 & 0  \\\\ 2 & 0  \\end{array}\\right] \\hspace{1cm}\n",
+    "B = \\left[ \\begin{array}{ccc} 4 & 0 \\\\ 5 & 0  \\end{array}\\right] \\hspace{1cm}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 5:** Find the **coordinate vector** of $F$ with respect to the basis $\\beta = \\{A,B,C,D\\}$ for $\\mathbb{M}_{2\\times 2}$.\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "A = \\left[ \\begin{array}{ccc} 1 & 0  \\\\ 0 & 1  \\end{array}\\right] \\hspace{1cm}\n",
+    "B = \\left[ \\begin{array}{ccc} 2 & 1  \\\\ 2 & 2  \\end{array}\\right] \\hspace{1cm}\n",
+    "C = \\left[ \\begin{array}{ccc} 3 & 0 \\\\ 1 & 4   \\end{array}\\right] \\hspace{1cm}\n",
+    "D = \\left[ \\begin{array}{ccc} 3 & 4\\\\ 1 & 1   \\end{array}\\right] \\hspace{1cm}\n",
+    "F = \\left[ \\begin{array}{ccc} 14 & 10\\\\ 7 & 11   \\end{array}\\right] \\hspace{1cm}\n",
+    "\\end{equation}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "**Exercise 6:** Let $\\mathbb{D}_{2\\times 2}$ be the set of $ 2 \\times 2 $ diagonal matrices. \n",
+    "\n",
+    "($a$)  Explain why $\\mathbb{D}_{2\\times 2}$ is a subspace of $\\mathbb{M}_{2\\times 2}$.\n",
+    "\n",
+    "($b$)  Find a basis for $\\mathbb{D}_{2\\times 2}$.\n",
+    "\n",
+    "($c$)  Determine the dimension of $\\mathbb{D}_{2\\times 2}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here"
+   ]
   }
  ],
  "metadata": {