Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
a1x1 + ....+ anxn = b ,
linear maps such as:
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}and their representations in vector spaces and through matrices.
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Topics Involved In Notes
- Functions of several variables
- Limits
- Partial Derivatives
Functions of several variables
We are interested in functions f from R n to R m (or more generally from a subset D ⊂ R n to R m called the domain of the function). A function f assigns to each x ∈ R n a point y ∈ R m and we write
y = f(x)
The set of all such points y is the range of the function.
Each component of y = (y1, . . . , ym) is real-valued function of x ∈ R n
written yi = f i(x)
and the function can also be written as the collection of n functions
If we also write out the components of x = (x1, . . . , xn), then are function can be written as m functions of n variables each:
y1 =f1(x1, . . . , xn)y2 =f2(x1, . . . , xn)
. . .
ym =fm(x1, . . . , xn)
The graph of the function is all pairs (x, y) with y = f(x) . It is a subset of Rn+m
limits
Consider a function y = f(x) from Rn to R m (or possibly a subset of R n ). Let x 0 = (x 0 1 , . . . xnn ) be a point in R n and let y 0 = (y 0 1 , . . . , y0 m ) be a point in R m. We say that y 0 is the limit of f as x goes to x 0 , written