Matrix applications and Newton's 2nd law

book/knowledge-library/classical-mechanics-1.ipynb

@@ -485,6 +485,50 @@
     "$$"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "858df39f",
+   "metadata": {},
+   "source": [
+    "Newton's 2nd law is often called the **equation of motion** for an object. To see why, notice that the law can be rewritten as:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e8bb9ef8",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "a = \\frac{F_{net}}{m}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "88ce133c",
+   "metadata": {},
+   "source": [
+    "And since acceleration is the second derivative of position, this can be written:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "09331495",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\frac{d^2 x}{dt^2} = \\frac{F_{net}}{m}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8620a61e",
+   "metadata": {},
+   "source": [
+    "Thus, solving for Newton's second law gives the function $x(t)$ that describes the motion of an object through space and time."
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "dae3e25a",

book/knowledge-library/linear-algebra.ipynb

@@ -1090,12 +1090,64 @@
     "Note that the new vector $\\vec v_2$ is equal to the original vector $\\vec v_1$ scaled by 2! So the matrix $S$ actually encodes the **linear transformation** of scaling by 2. In fact, all matrices can be thought of as _linear maps_ that map vectors onto their transformed versions. Common transformations, such as rotation, shearing, stretching, translation, and countless others can all be encoded through matrices! This is why matrices are useful!"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "id": "b54f07f0",
+   "metadata": {},
+   "source": [
+    "## Linearity"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1fb7d014",
+   "metadata": {},
+   "source": [
+    "Linear algebra might seem an unrelated mess of mathematical objects, problems, and techniques. But there is one theme underlying linear algebra - **linearity**. Anything that is linear is acted on by **linear operators**, to which the following rule applies:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "aeed4941",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "f(ax + by) = af(x) + bf(y)\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "2870e556",
+   "metadata": {},
+   "source": [
+    "For example, one linear operator would be vector addition, and another linear operator would be matrix multiplication - both follow this rule. Therefore vectors and matrices are linear as well."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "661426ef",
+   "metadata": {},
+   "source": [
+    "## What's the point?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "921c6cba",
+   "metadata": {},
+   "source": [
+    "Linear algebra has many arbitrary rules - but why learn them? What does linear algebra have to do with the real world, when it feels like you're just moving around a bunch of numbers in columns and rows? The answer is - a **lot**.\n",
+    "\n",
+    "For example, matrices are used to solve complicated systems of equations, and find optimal solutions to many problems. Vectors are used to model physical quantities, like force, position, and acceleration. Almost all computer graphics and machine learning uses both, and physics uses both in combination with other mathematics. In fact, the whole reason why linear algebra is called **linear** algebra is that it provides a consistent set of rules that apply to linear problems. Put it another way, any linear problem can be solved with linear algebra!"
+   ]
+  },
   {
    "cell_type": "markdown",
    "id": "8bdcb982",
    "metadata": {},
    "source": [
-    "## Transition to tensors"
+    "## Extra: motivation for tensors"
    ]
   },
   {
@@ -1175,7 +1227,7 @@
    "id": "a0031698",
    "metadata": {},
    "source": [
-    "This is where tensors come in. Tensors are generalizations of n-dimensional arrays - including vectors and matrices - which are invariant. This means, we can rewrite a scalar, vector, or matrix as an equivalent tensor that remains the same in whichever coordinate system we put it in. How cool is that! We'll explore tensors more in the next section."
+    "This is where tensors come in. Tensors are generalizations of n-dimensional arrays - including vectors and matrices - which are invariant. This means, we can rewrite a scalar, vector, or matrix as an equivalent tensor that remains the same in whichever coordinate system we put it in. How cool is that! We'll explore tensors some more a few sections later."
    ]
   }
  ],