017_LAlgIntro(Scala)

Import Notebook

ScaDaMaLe Course site and book

Linear Algebra Review / re-Introduction

by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

This is a breeze'y scalarific break-down of:

Using the above resources we'll provide a review of basic linear algebra concepts that will recur throughout the course. These concepts include:

  1. Matrices
  2. Vectors
  3. Arithmetic operations with vectors and matrices

We will see the accompanying Scala computations in the local or non-distributed setting.

Let's get a quick visual geometric interpretation for vectors, matrices and matrix-vector multiplications, eigen systems with real and comples Eigen values NOW from the following interactive visual-cognitive aid at:

Matrix by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

(watch now 0-61 seconds):

Linear Algebra Review by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

Breeze is a linear algebra package in Scala. First, let us import it as follows:

import breeze.linalg._
import breeze.linalg._

1. Matrix: creation and element-access

A matrix is a two-dimensional array.

Let us denote matrices via bold uppercase letters as follows:

For instance, the matrix below is denoted with A\mathbf{A}, a capital bold A.

A=(a11a12a13a21a22a23a31a32a33) \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}

We usually put commas between the row and column indexing sub-scripts, to make the possibly multi-digit indices distinguishable as follows:

A=(a1,1a1,2a1,3a2,1a2,2a2,3a3,1a3,2a3,3) \mathbf{A} = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{pmatrix}

  • Ai,j \mathbf{A}_{i,j} denotes the entry in ii-th row and jj-th column of the matrix A\mathbf{A}.
  • So for instance,
    • the first entry, the top left entry, is denoted by A1,1 \mathbf{A}_{1,1} .
    • And the entry in the third row and second column is denoted by A3,2 \mathbf{A}_{3,2} .
    • We say that a matrix with n rows and m columns is an nn by mm matrix and written as n×mn \times m
      • The matrix A\mathbf{A} shown above is a generic 3×33 \times 3 (pronounced 3-by-3) matrix.
      • And the matrix in Ameet's example in the video above, having 4 rows and 3 columns, is a 4 by 3 matrix.
    • If a matrix A\mathbf{A} is n×mn \times m , we write:
      • A∈Rn×m\mathbf{A} \in \mathbb{R}^{n \times m} and say that A\mathbf{A} is an R\mathbb{R} to the power of the n times m,
        • where, R\mathbb{R} here denotes the set of all real numbers in the line given by the open interval: (−∞,+∞) (-\infty,+\infty).

Let us created a matrix A as a val (that is immutable) in scala. The matrix we want to create is mathematically notated as follows:

A=(a1,1a1,2a1,3a2,1a2,2a2,3)=(123456) \mathbf{A} = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} 

val A = DenseMatrix((1, 2, 3), (4, 5, 6)) // let's create this 2 by 3 matrix
A: breeze.linalg.DenseMatrix[Int] = 1 2 3 4 5 6 
A.size
res0: Int = 6 
A.rows // number of rows
res1: Int = 2 
A.size / A.rows // num of columns
res2: Int = 3 
A.cols // also say
res3: Int = 3

Now, let's access the element a1,1a_{1,1}, i.e., the element from the first row and first column of A\mathbf{A}, which in our val A matrix is the integer of type Int equalling 1.

A(0, 0) // Remember elements are indexed by zero in scala
res4: Int = 1