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# Basic NumPy

## Problem

Given a matrix $A$ and a vector $\bb$, use NumPy to calculate the projection $A(A\trans A)^{-1}A\trans\bb$ and the solution $(A\trans A)^{-1}A\trans\bb$ of the least squares problem.

## Thought

NumPy is a Python package for handling arrays. You may use [Colab](https://colab.research.google.com/) or [Sage Cell](https://sagecell.sagemath.org/) to run Python code.

First, we have to import the package so that Python knows that you are going to use functions in the package.

import numpy as np

Here `np` is a shorthand for `numpy` , which can be any name, but we follow the convention from the [NumPy community](https://numpy.org/doc/stable/index.html).

Next, we have to construct an array (or a matrix).

A = np.array([
    [-1, -1],
    [1, 0], 
    [0, 1]
])

Here `A` is the name of your matrix, which can be a different name. In Python, a structure like `[1, 2, 3]` is called a `list` . Inside `np.array(...)` , we see the structure of `[ [...], ... [...] ]` , which is a list of lists. This is because a matrix can be viewed as a list of rows, and each row can be viewed as a list.

Finally, we introduce some functions.

1. matrix addision: `A + B`
2. matrix multiplication: `A.dot(B)`
3. transpose: `A.T`
4. inverse: `np.linalg.inv(A)`

## Sample answer

Run the SageMath code below or simply click [here](https://sagecell.sagemath.org/?q=okfbxa).

import numpy as np

A = np.array([
    [-1, -1],
    [1, 0], 
    [0, 1]
])
b = np.array([1,0,0])

ATAinv = np.linalg.inv(A.T.dot(A))
print("projection")
print(A.dot(ATAinv).dot(A.T).dot(b))
print("least square problem")
print(ATAinv.dot(A.T).dot(b))

*This note can be found at Course website > Learning resources.*