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Let f : R" × Rn → R be defined by f(x,y) = ã³ỹ. Verify that f is a bilinear(2-linear) form on Rn. That is, verify the following equalities: (a) For vectors, ỹ Є R^ and scalars c = R, ƒ (cx, y) = cƒ (x, y) (b) For vectors, ỹ Є R^ and scalars c = R, ƒ (ã, cỹ) = cf(x, y) (c) For vectors ‚ ÿ, žЄ R³, ƒ(˜+ ž‚ÿ) = f(x, y) + ƒ(¾‚ÿ) (d) For vectors, ÿ, žЄ R^, ƒ(˜, ÿ+ z) = ƒ (ï‚ÿ) + ƒ (x, z)

Let f : R" × Rn → R be defined by f(x,y) = ã³ỹ. Verify that f is a bilinear(2-linear) form on Rn. That is, verify the following equalities: (a) For vectors, ỹ Є R^ and scalars c = R, ƒ (cx, y) = cƒ (x, y) (b) For vectors, ỹ Є R^ and scalars c = R, ƒ (ã, cỹ) = cf(x, y) (c) For vectors ‚ ÿ, žЄ R³, ƒ(˜+ ž‚ÿ) = f(x, y) + ƒ(¾‚ÿ) (d) For vectors, ÿ, žЄ R^, ƒ(˜, ÿ+ z) = ƒ (ï‚ÿ) + ƒ (x, z)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let f : R R → R be defined by f(x,y) = xy. Verify that f is a bilinear(2-linear) form on Rn. That is, verify the following equalities: (a) For vectors x, y = R and scalars c = R, ƒ (cx, y) = cf(x, y) (b) For vectors x, y = Rn and scalars c = R, f(x, cy) = cf(x, y) (c) For vectors x, ÿ, žЄ R³, ƒ(ï+ž‚ÿ) = ƒ(ï‚ÿ) + ƒ(ž‚ÿ) (d) For vectors ‚ ÿ, žЄ Rª, ƒ(ï‚ÿ + z) = ƒ (x, ÿ) + ƒ (x, z)
Transcribed Image Text:Let f : R R → R be defined by f(x,y) = xy. Verify that f is a bilinear(2-linear) form on Rn. That is, verify the following equalities: (a) For vectors x, y = R and scalars c = R, ƒ (cx, y) = cf(x, y) (b) For vectors x, y = Rn and scalars c = R, f(x, cy) = cf(x, y) (c) For vectors x, ÿ, žЄ R³, ƒ(ï+ž‚ÿ) = ƒ(ï‚ÿ) + ƒ(ž‚ÿ) (d) For vectors ‚ ÿ, žЄ Rª, ƒ(ï‚ÿ + z) = ƒ (x, ÿ) + ƒ (x, z)
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