We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the multiplication of two vectors is and was hoping someone could help.
Note i have these completed in my understanding which may be wrong, so I'm just asking for some information about what I did do.
x = [0.5 0.5] y = [0.75, 1.25]
1) Multiply the following two vectors x y
2) Multiply the following two vectors xT y (transpose)
1)Are those two questions equivalent since x has both 0.5?
3)What is the dot product of x + w?
Is this just (0.5*0.75) + (0.5 * 1.25)?
Best Answer
Dot product has the form of $u_1v_1+u_2v_2+...u_nv_n$
In this case, you can write the standard form as $x_1y_1+x_2y_2+...x_ny_n$
Since, x = [0.5 0.5]
y = [0.75, 1.25]
$x_1 = 0.5 \space y_1 = 0.75$
$x_2 = 0.5 \space y_2 = 1.25$
Therefore, $x \cdot y$ is
$(0.5)(0.75) + (0.5)(1.25)$
$0.375 + 0.625 =1$
When we have to Transpose a Matrix, we need to change the order. For example, x = [0.5 0.5] is a 1 x 2 matrix because it has one row and two columns.
Therefore, $x^T$ would be a 2 x 1 matrix and would be written as $\left(\begin{array}{c} .5 \\ .5 \end{array}\right)$. We see two rows and one column and that's a good sign that we did it correctly.
For this problem, a $x^T$ form of a matrix is written as
$ \begin{array}{c} x_1 = 0.5\\ x_2 =0.5\\ \vdots\\ x_n \end{array}$
I've added the values so it would be easier to see.
Now $x^T$ multiply by $y$
$\left(\begin{array}{c} .5 \\ .5 \end{array}\right)\left(\begin{array}{cc} .75 & 1.25 \end{array}\right)$ would be $(0.5)(0.75) + (0.5)(1.25)$ = $0.375 + 0.625 = 1$