Understand array of vector notation

Question

I have the following notation:

$[p_1, p_2, p_3][\alpha_1 \lambda_1,\alpha_2 \lambda_2,\alpha_3 \lambda_3]^T$

where all $\alpha$ and $\lambda$ are scalars and all $p$'s are $3x1$ vectors.

How would the final result look like (i.e. what shape would it have)? My interpretation is that I perform a dot product between $p_i$ and $[\alpha_1 \lambda_1,\alpha_2 \lambda_2,\alpha_3 \lambda_3]^T$ and end up with a $1x3$ vector.

I am supposed to be able to add this result to a vector of the form $[x,y,z]^T$ but I can't see how it can have this form.

Answer 1

Hint:

You have: $$ \mathbf{p_i}=\begin{bmatrix} p_{i,1}\\p_{i,2}\\p_{i,3} \end{bmatrix} \quad i \in\{1,2,3\} $$ So: $$ [\mathbf{p_1},\mathbf{p_2},\mathbf{p_3}][\alpha_1 \lambda_1,\alpha_2 \lambda_2,\alpha_3 \lambda_3]^T= \begin{bmatrix} p_{1,1}&p_{2,1}&p_{3,1}\\ p_{1,2}&p_{2,2}&p_{3,2}\\ p_{1,3}&p_{2,3}&p_{3,3}\\ \end{bmatrix} \begin{bmatrix} \alpha_1\lambda_1\\ \alpha_2\lambda_2\\ \alpha_3\lambda_3\\ \end{bmatrix} $$

Answer 2

You have a $3 \times 3$ matrix multiplied with a $3 \times 1$ vector (note the transpose turns a $1 \times 3$ vector into a $3 \times 1$ vector). So the result is a $3 \times 1$ vector, namely $\alpha_1 \lambda_1 p_1 + \alpha_2 \lambda_2 p_3 + \alpha_3 \lambda_3 p_3$.