Consider the following vectors in R : That is, ----0--- V2 5 V = Equation Editor...

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Consider the following vectors in R³ : That is, ----0--- V2 5 V₁ = Equation Editor -5 1 T(v₁) 4v₁, T(v₂): = Levi defines a very particular linear map T: R³ R³ which • stretches the vector V₁ in the same direction by a factor of 4, compresses the vector v2 in the opposite direction by a factor of, and • compresses the vector V3 in the opposite direction by a factor of (a) Explain why T is well defined. (You may type v_1 or just v1 for v1, etc.) (b) Compute T4 (-8v₁ +6v2 +7v3). V3 = = -3 .:). -8 1 1 -V₂, T(√3)= - V3. 3 A- A I B I US X₂ X² Styles Font Size Words: 0 (c) Let A be the matrix of T (with respect to the standard basis in both domain and codomain) such that T(x) = Ax. Explain why A is diagonalisable. X =E ΩΣ Equation Editor (d) Help Levi find A. A = Equation G Q ΩΣ Editor 12 X == A- A I B IU S x₂ x² Styles Font ▼ Size (e) Having found A, Levi immediately writes down A¹ without performing any row operations or matrix multiplication. Explain why Levi was able to do this. 20 X DX GGG Q 13 *= 集 三 三 S ≡ Words: 0 ▼ Font A- A I B I US X₂ X² Styles ▼ Size Y & A Words: 0 Consider the following vectors in R³ : That is, ----0--- V2 5 V₁ = Equation Editor -5 1 T(v₁) 4v₁, T(v₂): = Levi defines a very particular linear map T: R³ R³ which • stretches the vector V₁ in the same direction by a factor of 4, compresses the vector v2 in the opposite direction by a factor of, and • compresses the vector V3 in the opposite direction by a factor of (a) Explain why T is well defined. (You may type v_1 or just v1 for v1, etc.) (b) Compute T4 (-8v₁ +6v2 +7v3). V3 = = -3 .:). -8 1 1 -V₂, T(√3)= - V3. 3 A- A I B I US X₂ X² Styles Font Size Words: 0 (c) Let A be the matrix of T (with respect to the standard basis in both domain and codomain) such that T(x) = Ax. Explain why A is diagonalisable. X =E ΩΣ Equation Editor (d) Help Levi find A. A = Equation G Q ΩΣ Editor 12 X == A- A I B IU S x₂ x² Styles Font ▼ Size (e) Having found A, Levi immediately writes down A¹ without performing any row operations or matrix multiplication. Explain why Levi was able to do this. 20 X DX GGG Q 13 *= 集 三 三 S ≡ Words: 0 ▼ Font A- A I B I US X₂ X² Styles ▼ Size Y & A Words: 0