Hi Guest!
Ok, this question is about matrix multiplicaton and the dot product (Also called: Inner product or scalar product).
u is a vector, so it has components: \(u_1,u_2,...u_n\)
Similarly, v is a vector, so it has components: \(v_1,v_2,...,v_n\)
Now, no need to actually make it in n-dimension. I mean we can, but why not simplify this and assume we are in 2 dimensional space?
That is, n=2
So, u \(= \) and v \(= \)
So our matrix A is: \(A=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\) (As given, u is the first row and v is the second row)
But we are given an angle and lengths of these vectors. Can we find the "dot product"?
Yes: \(u \dot {} v=u_1v_1+u_2v_2=cos(\theta)*\left || u |\right |*\left || v| \right |=cos(60)*3*2=3\)
So, we got: \(u_1v_1+u_2v_2=3\)
Next, let's see what the question really wants:
\(Au=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\) \(*\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\) = \(\begin{pmatrix} u_1^2+u_2^2\\ u_1v_1+u_2v_2 \end{pmatrix}\) = \(\begin{pmatrix} 3^2\\ 3 \end{pmatrix}=\begin{pmatrix} 9\\ 3 \end{pmatrix}\)
Now, in a similar way, find \(Av\).
Hope this helps!
