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I'm going through an article on linear regression, and they give the following formula for computing estimates:

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The convention is that all vectors are column vectors. So if n is the number of samples and m is the number of parameters, hw(x) has dimensions n x 1, wT has dimensions 1 x m, and x has dimensions n x m. The problem is that these dimensions don't work out for matrix multiplication.

Where have I gone wrong here?

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My guess is that $x$ here refers to a single datapoint with shape $m$x$1$. So $h_w(x)$ and $w^Tx$ both have shape $1$. In the case of multiple datapoints given as a matrix $X$ of shape $n$x$m$, the equation becomes:

$h_w(X) = Xw$

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  • $\begingroup$ Indeed I think would be more consistent to have written the case in the OP as $x^Tw$. It doesn't make much sense to think of one data point as a column vector when the design matrix is inherently by convention a bunch of data as row vectors. $\endgroup$
    – Sean Owen
    Mar 2, 2020 at 3:01

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