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Engineering Calculus Notes 429

# Engineering Calculus Notes 429 - −→ k(Figure 4.2 It is...

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4.1. LINEAR MAPPINGS 417 There is a more geometric characterization of linear mappings which is often useful: Remark 4.1.2. A mapping L : R 3 R 3 is linear if and only if it preserves linear combinations: that is, for any two vectors −→ v and −→ v and any two scalars α and β , L ( α −→ x + β −→ v ) = αL ( −→ v ) + βL ( −→ v ) . Geometrically, this means that the image under L of any triangle (with vertex at the origin) is again a triangle (with vertex at the origin). As an example, consider the mapping P : R 3 R 3 that takes each vector −→ x to its perpendicular projection onto the plane x + y + z = 0 through the origin with normal vector −→ n = −→ ı + −→
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Unformatted text preview: + −→ k (Figure 4.2 ). It is x y z −→ n −→ x P ( −→ x ) Figure 4.2: Projection onto a Plane geometrically clear that this takes triangles through the origin to triangles through the origin, and hence is linear. Since any vector is the sum of its projection on the plane and its projection on the normal line, we know that P can be calculated from the formula P ( −→ x ) = −→ x − proj −→ u −→ x = −→ x − ( −→ x · −→ u ) −→ u , where −→ u = −→ n / √ 3 is the unit vector in the direction of...
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