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A little silly of a question, but I'm a little confused on the difference between a scalar sum and a vector sum.

So I'm taking a proof-based sampling course and the textbook defined a design linear estimator for a vector of values $(y_1, y_2, \cdots, y_N)$ as \begin{align} \hat{\theta} &= \sum_{i \in A}{w_iy_i}. \end{align}

But then they specified that they stated this theorem for scalars, although it works for vectors as below: \begin{align} \boldsymbol{\hat{\theta}} = \sum_{i \in A}{w_i\mathbf{y}_i}, \end{align} where they defined $\mathbf{y_i}$ as a column vector.

Again, I know this is a really silly question, but I'm struggling to find the difference between the two lol, other than the fact that we would have a vector as a result versus a scalar.

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    $\begingroup$ The difference between the two is your own last sentence. $\endgroup$
    – Randall
    Aug 26, 2022 at 16:15
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    $\begingroup$ @Randall oh man I guess you really shouldn't doubt yourself, lol. $\endgroup$
    – JerBear
    Aug 26, 2022 at 16:19

1 Answer 1

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As noticed in the comments, the first one is a scalar obtained as a sum of numbers:

$$\hat{\theta} = \sum_{i \in A}{w_iy_i}=w_1y_1+\ldots w_ny_n$$

the second one is a vector obtained as sum of vectors, that is explicitly

$$\boldsymbol{\hat{\theta}} = \sum_{i \in A}{w_i\mathbf{y}_i}=w_1\mathbf{y}_1+\ldots +w_n\mathbf{y}_n =\\=w_1\begin{pmatrix}y_{11}\\\vdots\\y_{m1} \end{pmatrix}+\ldots+w_n \begin{pmatrix}y_{1n}\\\vdots\\y_{mn} \end{pmatrix}=\begin{pmatrix}w_1y_{11}+\ldots+w_ny_{1n}\\\vdots\\w_1y_{m1}+\ldots+w_ny_{mn} \end{pmatrix}$$

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  • $\begingroup$ Wait but doesn't this mean that $y$ is an $m \times n$ matrix? In my textbook they only specified that $y$ is a column vector $\endgroup$
    – JerBear
    Aug 26, 2022 at 16:47
  • $\begingroup$ @JerBear Each $\mathbf{y}_i$ is a column vector, here I'm assuming we have $n$ of them each one with $m$ components. I've added another line to be more clear. $\endgroup$
    – user
    Aug 26, 2022 at 16:52
  • $\begingroup$ You are welcome! Bye $\endgroup$
    – user
    Aug 26, 2022 at 16:55

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