Page 378 - 35Linear Algebra
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378 Movie Scripts
In fact this is a system of linear equations whose solutions form a plane with
normal vector (1, 2, 5). As an augmented matrix the system is simply
1 2 5 3 .
This is actually RREF! So we can let x be our pivot variable and y, z be
represented by free parameters λ 1 and λ 2 :
x = λ 1 , y = λ 2 .
Thus we write the solution as
x = −2λ 1 −5λ 2 +3
y = λ 1
z = λ 2
or in vector notation
x 3 −2 −5
y = 0 + λ 1 1 + λ 2 0 .
z 0 0 1
This describes a plane parametric equation. Planes are ‘‘two-dimensional’’
because they are described by two free variables. Here’s a picture of the
resulting plane:
The Story of Your Life
This video talks about the weird notion of a ‘‘length-squared’’ for a vector
2 2 2
v = (x, t) given by ||v|| = x − t used in Einstein’s theory of relativity. The
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