Key Concepts

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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
  1. 1
    Linear 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…
  2. 2
    The fundamental building blocks of linear algebra are vectors. Vectors are defined as quantities having both direction and magnitude, compared to scalar quantities that only have magnitude. In …
  3. 3
    Now 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…
  4. 4
    An 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…
  5. 5
    Now that we’ve learned about vector quantities, we can expand upon those lessons and focus on matrices. A matrix is a quantity with m rows and n columns of data. For example, we can combine m…
  6. 6
    Like 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…
  7. 7
    There 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…
  8. 8
    An 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_…
  9. 9
    Now 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 …
  10. 10
    The 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…
  11. 11
    In this lesson, we learned about the fundamental building blocks of linear algebra: vectors and matrices, their important operations, and how to use them to solve important problems. We discussed …

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