Andrew Thangaraj

Aug-Nov 2020

- 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 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

- Express relationships between quantities and understand underlying mechanisms
- 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

\(V,W\): Two vector spaces

A *linear map* from \(V\) to \(W\) is a function \(T:V\to W\) satisfying the following two properties:

**Additivity**: \(T(u+v) = Tu + Tv\) for \(u,v\in V\)**Homogeneity**: \(T(\lambda v) = \lambda Tv\) for \(\lambda\in F\), \(v\in V\)

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

**Linear maps preserve linear combinations**

*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

- \(0(v)=0\) or \(0v=0\) for all \(v\in V\)
**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

\(T(x)=4x\) 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 \(F^3\to F^3\)

Suppose \(T:V\to W\) is a linear map and \(V\) is finite-dimensional with a basis \(\{v_1,v_2,\ldots,v_n\}\).

\(T\) is fully defined by specifying \(T(v_1),T(v_2),\ldots,T(v_n)\).

*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 \(T(v_i)=w_i\).

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 \[\begin{bmatrix} A_{11}&A_{12}&\cdots&A_{1n}\\ A_{21}&A_{22}&\cdots&A_{2n}\\ \vdots&\vdots&\vdots&\vdots\\ A_{m1}&A_{m2}&\cdots&A_{mn} \end{bmatrix}.\]

\(T(x)=4x\) from \(F\to F\)

- \(T(1)=4\), Matrix: \([4]\)

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

\(T(1,0)=3\), \(T(0,1)=4\).

Matrix: \([3\ \ 4]\).

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

\(T(1,0)=(3,5)\), \(T(0,1)=(4,-7)\).

Matrix: \(\begin{bmatrix}3&4\\5&-7\end{bmatrix}\).

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

\(T(1,0,0)=(3,5,9)\), \(T(0,1,0)=(4,-7,2)\), \(T(0,0,1)=(1,-2,-4)\).

Matrix: \(\begin{bmatrix}3&4&1\\5&-7&-2\\9&2&-4\end{bmatrix}\).

*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

*Model*

- Incidence of flu in people who already got flu in the previous year: 10%
- Incidence of flu in people who have not gotten flu in the previous year: 30%
- Overall population change is insignificant

*Equations*

\(\begin{align} x_2 &= 0.1 x_1 + 0.3 y_1\\ y_2 &= 0.9 x_1 + 0.7 y_1 \end{align}\)

*Linear map and matrix*

\(T(x,y)=(0.1x+0.3y,0.9x+0.7y)\)

\(\begin{bmatrix} 0.1&0.3\\ 0.9&0.7 \end{bmatrix}\)