Linear Maps and Matrices

Recap

• Vector space \(V\) over a scalar field \(F\)
  • \(F\): real field \(\mathbb{R}\) or complex field \(\mathbb{C}\) in this course
• Linear combinations
  • \(a_1v_1+a_2v_2+\cdots\) for \(v_i\in V\) and \(a_i\in F\)
• Span\((v_1,\ldots,v_n)\)
  • All linear combinations
• Subspace
  • Subset closed under linear combinations
• Linearly independent set of vectors
  • No non-trivial linear combination is zero
• Sums, direct sums
  • Given a subspace \(U\), there is a subspace \(W\) such that \(V=U\oplus W\)
• Gaussian elimination
  • Find linear dependence by reducing a set of vectors to echelon form

Functions or maps

• Functions or maps from real numbers to real numbers
  • Express relationships between quantities and understand underlying mechanisms
    • Linear function: one quantity increases proportionally with another
    • Quadratic, polynomial, exponential, logarithmic, trigonometric etc.
  • Most common and simplest starting point is a linear function
• Functions or maps from vectors to vectors
  • Express and understand relationships between multiple related physical quantities
  • Linear maps: most important class and simplest starting point
• This course: we focus on algebra of linear maps
  • Very little (or no) emphasis on physical aspects of the maps
  • We will study how linear maps operate on vectors and try to simplify/classify them

Linear maps

For \(a,b\in F\) and \(u,v\in V\), \[T(au+bv) = aT(u)+bT(v)\]

Linear maps preserve linear combinations

Examples of linear maps

Trivial, but important

• zero, denoted \(0\), from \(V\) to \(W\)
  • \(0(v)=0\) or \(0v=0\) for all \(v\in V\)
    • \(0\) denotes multiple things in the definition above
• identity, denoted \(1\), from \(V\) to \(V\)
  • \(1(v)=v\) or \(1v=v\) for all \(v\in V\)

Polynomials: \(P(\mathbb{R})\), denotes polynomials with real coefficients

• differentiation, denoted \(D\), from \(P(\mathbb{R})\to P(\mathbb{R})\)
  • \(D(p(x)) = p'(x)\), which is the usual derivative
  • Why linear? \((p+q)'=p'+q'\) and \((\lambda p)'=\lambda p'\)
• integration, defined as \(Tp=\int_0^1 p(x)dx\), from \(P(\mathbb{R})\to\mathbb{R}\)
  • Integration is linear
• multiplication by \(x^2\), defined as \((Tp)(x)=x^2 p(x)\), from \(P(\mathbb{R})\to P(\mathbb{R})\)
  • Check linearity

Maps from \(F^n\to F^m\)

• \(T(x)=4x\) from \(F\to F\)

• \(T(x)=4x+3\) from \(F\to F\)

• \(T(x,y)=3x+4y\) from \(F^2\to F\)

• \(T(x,y)=(3x+4y,5x-7y)\) from \(F^2\to F^2\)

• \(T(x,y)=(3x+4y+5xy,5x-7y+2)\) from \(F^2\to F^2\)

• \(T(x,y,z)=(3x+4y+z,5x-7y-2z,9x+2y-4z)\) from \(F^3\to F^3\)

Linear maps and basis

Proof

• Any \(v\in V\) can be written as \(v=a_1v_1+\cdots+a_nv_n\), \(a_i\in F\)
• So, \(T(v)=a_1T(v_1)+\cdots+a_nT(v_n)\)

In other words, for any \(n\) vectors \(w_1,\ldots,w_n\in W\) and a basis \(\{v_1,v_2,\ldots,v_n\}\) of \(V\), there is a linear map \(T:V\to W\) such that \(T(v_i)=w_i\).

Matrix representation of a linear map

Suppose \(T:V\to W\) is a linear map, \(V\) is finite-dimensional with a basis \(\{v_1,\ldots,v_n\}\) and \(W\) is finite-dimensional with a basis \(\{w_1,\ldots,w_m\}\).

Let \(T(v_j)=A_{1j}w_1+\cdots+A_{mj}w_m\), where \(A_{ij}\in F\) for \(i=1,\ldots,m\), \(j=1,\ldots,n\). The scalars \(A_{ij}\) fully specify the linear map \(T\) under the given bases for \(V\) and \(W\).

They are written in matrix form as \[\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ A_{m1}&A_{m2}&\cdots&A_{mn} \end{bmatrix}.\]

Examples: \(F^n\to F^m\), standard bases

Simple modelling example

Variables

• \(x_1\): number of people in India who got flu in 2020
• \(y_1\): number of people in India who did not get flu in 2020
• \(x_2\): number of people in India who will get flu in 2021
• \(y_2\): number of people in India who will not get flu in 2021

Quiz