Exercises and solutions

Author: bvanderlei

Applications.ipynb

@@ -267,6 +267,15 @@
     "$(b)$ Expand your work in part $(a)$ to a seventh degree poynomial that passes through all eight planets. The first object in the Kuiper Belt, Ceres, was discovered by Giuseppe Piazzi in 1801. Ceres has an average distance from the sun of 413.5 million km. Based on the points on the graph, estimate the orbital velocity of Ceres. What does the polynomial suggest the value would be?"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "## Code solution here."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},

Inner_Products.ipynb

@@ -6,7 +6,7 @@
    "source": [
     "## Inner Products\n",
     "\n",
-    "In this chapter, we introduce the idea of an inner product, examine the additional structure that it imposes on  a vector space, and look at some common applications.  An inner product is an operation on two vectors in a vector space that is defined in such a way as to satisfy certain algebraic requirements.  To begin, we will focus only on one specific inner product defined for vectors in  $\\mathbb{R}^n$.  Later in the chapter we will consider other examples of inner products in $\\mathbb{R}^n$.\n",
+    "In this chapter we introduce the idea of an inner product, examine the additional structure that it imposes on  a vector space, and look at some common applications.  An inner product is an operation on two vectors in a vector space that is defined in such a way as to satisfy certain algebraic requirements.  To begin, we will focus only on one specific inner product defined for vectors in  $\\mathbb{R}^n$.  Later in the chapter we will consider other examples of inner products in $\\mathbb{R}^n$.\n",
     "\n",
     "The **dot product** is the most common example of an inner product.  For vectors $U$ and $V$ in $\\mathbb{R}^n$, the dot product is written as $U\\cdot V$, and is defined in the following way.\n",
     "\n",
@@ -445,7 +445,7 @@
    "source": [
     "### Exercises\n",
     "\n",
-    "- Create a Python function named $\\texttt{Magnitude}$ that accepts as an argument vector in the form of a NumPy array object with shape $n\\times 1$, and returns the magnitude of that vector."
+    "**Exercise 1:** Create a Python function named $\\texttt{Magnitude}$ that accepts as an argument vector in the form of a NumPy array object with shape $n\\times 1$, and returns the magnitude of that vector."
    ]
   },
   {
@@ -461,12 +461,12 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "- Use the $\\texttt{random}$ module of NumPy to generate random $4\\times 1$ vectors $U$, $V$, and $W$, and a random scalar $k$.  Verify the algebraic properties of the dot product listed above.  Try to make use of a conditional statement to check each statement."
+    "**Exercise 2:** Use the $\\texttt{random}$ module of NumPy to generate random $4\\times 1$ vectors $U$, $V$, and $W$, and a random scalar $k$.  Verify the algebraic properties of the dot product listed above.  Try to make use of a conditional statement to check each statement."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -477,7 +477,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "- Let $X$ and $Y$ be the following vectors.\n",
+    "**Exercise 3:** Let $X$ and $Y$ be the following vectors.\n",
     "\n",
     "$$\n",
     "\\begin{equation}\n",
@@ -486,7 +486,7 @@
     "\\end{equation}\n",
     "$$\n",
     "\n",
-    "- Determine the angle between $X$ and $Y$.  (*You will need the $\\texttt{acos}$ function in Python's $\\texttt{math}$ module.*)"
+    "$(a)$ Determine the angle between $X$ and $Y$.  (*You will need the $\\texttt{acos}$ function in Python's $\\texttt{math}$ module.*)"
    ]
   },
   {
@@ -502,7 +502,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "- Find a vector in $\\mathbb{R}^3$ that is orthogonal to the vector $X$.  Verify your answer with a computation."
+    "$(b)$ Find a vector in $\\mathbb{R}^3$ that is orthogonal to the vector $X$.  Verify your answer with a computation."
    ]
   },
   {
@@ -518,7 +518,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "- Construct a unit vector $Z$ such that $Z\\cdot Y = -||Y||$.  Verify your answer with a computation."
+    "$(c)$ Construct a unit vector $Z$ such that $Z\\cdot Y = -||Y||$.  Verify your answer with a computation."
    ]
   },
   {