BIZNIQUE / LINEAR ALGEBRA / 09
See a matrix act on a grid
See a matrix act on a grid
The transformed basis vectors tell you where every other vector will go.
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THE IDEA
Read the example.
Read the example.
The first and second columns of a two-dimensional matrix are the images of the two basis vectors. Once those two arrows are known, linearity determines the transformation of the entire grid.
A(x₁e₁ + x₂e₂) = x₁Ae₁ + x₂Ae₂
Use it well
Use it well
A shear can preserve area while changing lengths and angles. Preserving one geometric property does not imply that all geometry stays the same.
TRY IT YOURSELF
A quick check.
A quick check.
For A = [[1, 2], [0, 1]], identify Ae₁ and Ae₂.
Check your reasoning
Check your reasoning
The columns give Ae₁ = (1, 0) and Ae₂ = (2, 1).
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