Problem II Consider a linear operator L: R4¹ → R4 defined by the formula below, where V₁ = (1, 1, 1, 1), v₂ = (1, 1, 0, 0), V3 (1, 2, 0, -1) and v4 = (0, 0, 1, 1) (the formula involves the dot product and scalar multiplication). = Find a matrix M such that L(u) = Mu for every u € Rª, where u and L(u) are regarded as column vectors. • V1 2 - 21. L(u) = (u V₁)V₂ + (U · V3)V4. 22. L(u) = (u · v₁)V2 — (U · V3)V4. 23. L(u) = (u · V₁)V₁ + (U · V3) V2. 24. L(u) = (u · v₁)V4 — (U · V3)V2. 25. L(u) = (u V₂)V₁ + (U · V3)V4. 26. L(u) = (u · v2)V₁ — (U · V3)V4. 27. L(u) = (u V₁)V2 + (U · V4)V3. 28. L(u) = (u · v1)V2 — (U · V4)V3. 29. L(u) = (u · V2)V3 + (U · V₁)V4. 30. L(u) = (u · v₂)V3 – (U · V₁)V4. . 1 31. L(u) = (u · V₁)V₁ + (U · V₂)V₂2 + (V3 · V4)U. 32. L(u) = (u · V₁)V2 + (U · V2)V₁ + (V3 · V4)U. 33. L(u) = (u · V₂)V2 + (U · V3)V3 + (V₁ · V4)U. 34. L(u) = (u · V₂)V3 + (U · V3)V2 + (V₁ · V4)U. 35. L(u) = (u V₁)V₁ + (U · V₁)V4 + (V2 · V3)U. 36. L(u) = (u · V₁)V₁ + (U · V4)V₁ + (V₂ · V3)U. 37. L(u) = (u · v₂)V2 + (U · V4)V4 + (V₁ · V3)U. 38. L(u) = (u · v2)V4 + (U · V4)V2 + (V₁ · V3)U. 39. L(u) = (u V3)V3 + (U · V4)V4 + (V₁ · V₂)u. 40. L(u) = (u · V3)V4 + (U · V4)V3 + (V₁ · V2)U.
Problem II Consider a linear operator L: R4¹ → R4 defined by the formula below, where V₁ = (1, 1, 1, 1), v₂ = (1, 1, 0, 0), V3 (1, 2, 0, -1) and v4 = (0, 0, 1, 1) (the formula involves the dot product and scalar multiplication). = Find a matrix M such that L(u) = Mu for every u € Rª, where u and L(u) are regarded as column vectors. • V1 2 - 21. L(u) = (u V₁)V₂ + (U · V3)V4. 22. L(u) = (u · v₁)V2 — (U · V3)V4. 23. L(u) = (u · V₁)V₁ + (U · V3) V2. 24. L(u) = (u · v₁)V4 — (U · V3)V2. 25. L(u) = (u V₂)V₁ + (U · V3)V4. 26. L(u) = (u · v2)V₁ — (U · V3)V4. 27. L(u) = (u V₁)V2 + (U · V4)V3. 28. L(u) = (u · v1)V2 — (U · V4)V3. 29. L(u) = (u · V2)V3 + (U · V₁)V4. 30. L(u) = (u · v₂)V3 – (U · V₁)V4. . 1 31. L(u) = (u · V₁)V₁ + (U · V₂)V₂2 + (V3 · V4)U. 32. L(u) = (u · V₁)V2 + (U · V2)V₁ + (V3 · V4)U. 33. L(u) = (u · V₂)V2 + (U · V3)V3 + (V₁ · V4)U. 34. L(u) = (u · V₂)V3 + (U · V3)V2 + (V₁ · V4)U. 35. L(u) = (u V₁)V₁ + (U · V₁)V4 + (V2 · V3)U. 36. L(u) = (u · V₁)V₁ + (U · V4)V₁ + (V₂ · V3)U. 37. L(u) = (u · v₂)V2 + (U · V4)V4 + (V₁ · V3)U. 38. L(u) = (u · v2)V4 + (U · V4)V2 + (V₁ · V3)U. 39. L(u) = (u V3)V3 + (U · V4)V4 + (V₁ · V₂)u. 40. L(u) = (u · V3)V4 + (U · V4)V3 + (V₁ · V2)U.
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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Parts 21-24 Please
Transcribed Image Text:**Problem II** Consider a linear operator \( L : \mathbb{R}^4 \rightarrow \mathbb{R}^4 \) defined by the formula below, where \( \mathbf{v}_1 = (1, 1, 1, 1), \mathbf{v}_2 = (1, 1, 0, 0), \mathbf{v}_3 = (1, 2, 0, -1) \) and \( \mathbf{v}_4 = (0, 0, 1, 1) \) (the formula involves the dot product and scalar multiplication).
Find a matrix \( M \) such that \( L(\mathbf{u}) = M\mathbf{u} \) for every \( \mathbf{u} \in \mathbb{R}^4 \), where \( \mathbf{u} \) and \( L(\mathbf{u}) \) are regarded as column vectors.
21. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_1)\mathbf{v}_2 + (\mathbf{u} \cdot \mathbf{v}_3)\mathbf{v}_4 \).
22. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_1)\mathbf{v}_2 - (\mathbf{u} \cdot \mathbf{v}_3)\mathbf{v}_4 \).
23. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_1)\mathbf{v}_4 + (\mathbf{u} \cdot \mathbf{v}_3)\mathbf{v}_2 \).
24. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_1)\mathbf{v}_4 - (\mathbf{u} \cdot \mathbf{v}_3)\mathbf{v}_2 \).
25. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_2)\mathbf{v}_1 + (\mathbf{u} \cdot \mathbf{v}_3)\mathbf{v}_4 \).
26. \( L(\mathbf{u}) = (\mathbf{u} \cdot \mathbf{v}_2)\mathbf{v}_1 - (\mathbf
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