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title: "14 - Linear Algebra"
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## Motivation

R provides out-of-box support for matrices and vectors. This is not the case for C++ code, where you need to use a library such as [Armadillo](https://arma.sourceforge.net/) that has a wrapper for R [@vargassepulveda25]. One of the nicest Armadillo features is that it has a similar syntax to MATLAB.

## Using Armadillo

[cpp11armadillo](https://pacha.dev/cpp11armadillo/) provides a detailed R-specific documentation with worked examples.

Because the examples in the cpp11armadillo are self-contained and with detailed explanations, this vignette will not repeat them here.

Here is one of the over five-hundred examples from the cpp11armadillo package to compute $\hat{\beta} = (X^TX)^{-1}X^Ty$ for a linear model using the cpp4r concepts already covered:

```cpp
Mat<double> ols_(const doubles_matrix<>& y, const doubles_matrix<>& x) {
  Mat<double> Y = as_Mat(y);  // Col<double> Y = as_Col(y); also works
  Mat<double> X = as_Mat(x);

  Mat<double> XtX = X.t() * X;             // X'X
  Mat<double> XtX_inv = inv(XtX);          // (X'X)^(-1)
  Mat<double> beta = XtX_inv * X.t() * Y;  // (X'X)^(-1)(X'Y)

  return beta;
}

[[cpp4r::register]] doubles_matrix<> ols_mat_(const doubles_matrix<>& y,
                                              const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles_matrix(beta);
}

[[cpp4r::register]] doubles ols_dbl_(const doubles_matrix<>& y,
                                     const doubles_matrix<>& x) {
  Mat<double> beta = ols_(y, x);
  return as_doubles(beta);
}
```

One of the advantages of cpp11armadillo is that it connects cpp11 (and cpp4r) with Armadillo, allowing you to use the
`as_Mat()` and `as_doubles_matrix()` functions as the "bread" of the sandwich, where pure Armadillo code serves as the
"meat".

## References