Page 22 - 35Linear Algebra
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22 What is Linear Algebra?
Writing our fruity equations as an equality between 2-vectors and then using
these rules we have:
2 x + 6 y = 20 2x + 6y 20 2 6 20
⇐⇒ = ⇐⇒ x +y = .
4 x + 8 y = 28 4x + 8y 28 4 8 28
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Now we introduce a function which takes in 2-vectors and gives out 2-vectors.
We denote it by an array of numbers called a matrix .
2 6 2 6 x 2 6
The function is defined by := x + y .
4 8 4 8 y 4 8
A similar definition applies to matrices with different numbers and sizes.
Example 6 (A bigger matrix)
x
1 0 3 4 1 0 3 4
y
5 0 3 4 := x 5 + y 0 + z 3 + w 4 .
z
−1 6 2 5 −1 6 2 5
w
Viewed as a machine that inputs and outputs 2-vectors, our 2 × 2 matrix
does the following:
x 2x + 6y
.
y 4x + 8y
Our fruity problem is now rather concise.
Example 7 (This time in purely mathematical language):
x 2 6 x 20
What vector satisfies = ?
y 4 8 y 28
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To be clear, we will use the term 2-vector to refer to stacks of two numbers such
7
2
3
as . If we wanted to refer to the vectors x + 1 and x − 1 (recall that polynomials
11
3
2
are vectors) we would say “consider the two vectors x − 1 and x + 1”. We apologize
through giggles for the possibility of the phrase “two 2-vectors.”
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