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diff --git a/content/posts/linear_algebra.md b/content/posts/linear_algebra.md index 0be65f9..5239a97 100644 --- a/content/posts/linear_algebra.md +++ b/content/posts/linear_algebra.md @@ -18,4 +18,16 @@ $$ \mathbf{y} = c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2 + c_3 \mathbf{x}_3 $$ ## Matrices
-A matrix is a useful way of representing a transformation. A matrix is a set of vector as described in the earlier section stacked together horizontally. Another way of thinking of a matrix is as a transformation of basis. In 3 dimensions, the unit vector $x$ axis is defined by $\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$
\ No newline at end of file +A matrix is a useful way of representing a transformation. A matrix is a set of vector as described in the earlier section stacked together horizontally. Another way of thinking of a matrix is the transformation of reference axes. In 3 dimensions, the unit vector along the $x$, $y$ and $z$ axis is defined as $\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$, $\begin{bmatrix} 0 & 1 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$. If we stack these one top of each other, we arrive at the identity matrix or the default axes. We can transform these axes to point to any other directions by just stacking those vectors into a matrix and then multiplying them. Notice that the operation carried out is now just the dot product of the vector with axes that was just defined. The dot product is just a measure of how much the vector is pointing in the direction of another vector which in our case is the new axes.
+
+$$ \mathbf{v} = \begin{bmatrix} 1 & 2 & 3\\\\ 3 & 5 & 8 \\\\ 11 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\\\ y \\\\ z \end{bmatrix} = \begin{bmatrix} x + 2y + 3z \\\\ 3x + 5y + 8z \\\\ 11x + 3y + z \end{bmatrix} $$
+
+This operation is another way of representing the linear transformation discussed in the previous section. I like to this of this operation as refocusing any and all vectors to a different section of space. This can be generalized with a change in the origin as well using the affine transformation written as:
+
+$$ \mathbf{y} = \mathbf{A} \mathbf{x} + \mathbf{b} $$
+
+where $\mathbf{A}$ is the change in perspective and $\mathbf{b}$ is the change in point of reference.
+
+## Matrix Rank
+
+This section is to be continued.
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