# Linear Algebra

Learn about the fundamentals of linear algebra!

Start## Key Concepts

Review core concepts you need to learn to master this subject

Scalars, Vectors, and Matrices

Matrix Addition

Matrix Multiplication

Matrix Transpose

Vector Magnitude

Basic Vector Operations

Dot Product

Augmented Matrix

Scalars, Vectors, and Matrices

Scalars, Vectors, and Matrices

```
import numpy
# Scalar code example
x = 5
# Vector code example
x = numpy.array([1, 2, 3, 4])
# Matrix code example
x = numpy.array([1, 2, 3], [4, 5, 6], [7, 8, 9])
```

*Scalars*, *vectors*, and *matrices* are fundamental structures of linear algebra, and understanding them is integral to unlock the concepts of deep learning.

- A scalar is a singular quantity like a number.
- A vector is an array of numbers (scalar values).
- A matrix is a grid of information with rows and columns.

They can all be represented in Python using the NumPy library.

Introduction to Linear Algebra

Lesson 1 of 2

- 1Linear algebra is a branch of mathematics that studies linear transformations, including the use of vectors and matrices. Linear algebra enables us to solve problems like linear regression, as well…
- 3Now that we know what a vector is and how to represent one, we can begin to perform various operations on vectors and between different vectors. As we’ve previously discussed, the basis of linear a…
- 4An important vector operation in linear algebra is the
*dot product*. A dot product takes two equal dimension vectors and returns a single scalar value by summing the products of the vectors’ corre… - 6Like with vectors, there are fundamental operations we can perform on matrices that enable the linear transformations needed for linear algebra. We can again both multiply entire matrices by a scal…
- 7There are a couple of important matrices that are worth discussing on their own. #### Identity matrix The
*identity matrix*is a square matrix of elements equal to 0 except for the elements along… - 8An extremely useful application of matrices is for solving systems of linear equations. Consider the following system of equations in its algebraic form. \begin{aligned} a_1 x + b_1 y + c_1 z = d_…
- 9Now that we have our system of linear equations in augmented matrix form, we can solve for the unknown variables using a technique called
*Gauss-Jordan Elimination*. In regular algebra, we may try … - 10The
*inverse of a matrix*, *A -1 *, is one where the following equation is true: AA^{-1} = A^{-1}A = I This means that the product of a matrix and its inverse (in either order) is equal to the ide…

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