Introduction to Multi-Var Calc and Lin Alg

Published on: Wed Sep 01 2010

Previously in calculus we did things like y=mx+b with m being the slope and for x not varying much.

The goal of this class is to understand things like z = x2+y2, a three dimensional shape, similar to a bowl. One dimensional curves in space and vector fields, like the flow of water in a river.

The best way to do this is through Vectors and linear transformations (Matrices).

The right hand coordinate system, has x emerging from the paper, y traveling to the right and z to the top of the page. The left hand coordinate system swaps y and x.

A vector has length and direction. For this class we will write in notation R3 meaning the real number system in three dimensions. We can describe locations in Rn by position vectors. Points are different than vectors. You can't add points, but it vectors can be added or subtracted.

For the purposes of this class [#,#] is a vector Square Brackets, (0,0) is a point.

There is a special vector, the zero vector [0,0]. The length of vector v is the absolute value of v. I should write little hats on top of my vectors, but I can't easily type those in.

Describe a Line in a plane
At some time t, the position of the vector passing through (0,0) of the plane and with a slope of one, the line can be described as t[1,1]

For some other line with a slope of one-half and passing through the point (0,2) this line can be described as [ 2t , t+2 ]

Line = starting point + t(direction)
Line = u +tv
and v are vectors. t is not. u is the zero vector for this point.

To describe a plane, use two vectors
[1,0,0] + t[-1,1,0]+s[-1,0,1]

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