3. Let L: R4 → R³ be defined by1U211U2L12 1Let S and T be the naturalU3Из1-2 1U4U4bases for R4 and R3, respectively and consider the ordered bases1S' =1and T' =1for R and R3, respectively. Find the1representation of L with respect to (a) S and T: (b) S' and T'.

Answered: Image /qna-images/answer/6fe8b8ba-1627-463a-ab82-ef63f78b0b61.jpg

Answered: 3. Let L: R4 → R³ be defined by 1 U2 1…

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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We define a inner product in = U₁V₁ + 2U₂V₂ + U3V3 in R³ for u = (U₁, U₂, U3) and v= (V₁, V₂, V3) members of R³. We also define S basis R³. S= = {W₁, W₂, W3} where w₁ = : (1,1,1), W₂ = (1,1,0), W3 = (1,0,0) a) Determine and explain whether the basis S is an orthonormal basis with the given inner product? b) If S is not an orthonormal basis, then determine the orthonormal basis S' (for the final step of normalization, if the number in the vector is difficult to simplify then just leave it in its initial form).

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3. Let L: R4 → R³ be defined by 1 U2 1 1 U2 L 1 2 1 Let S and T be the natural U3 Из 1 -2 1 U4 U4 bases for R4 and R3, respectively and consider the ordered bases 1 S' = 1 and T' = 1 for R and R3, respectively. Find the 1 representation of L with respect to (a) S and T: (b) S' and T'.