Consider the following vectors in R : That is, ----0--- V2 5 V = Equation Editor -5 1 T(v) 4v, T(v): = Levi defines
Consider the following vectors in R : That is, ----0--- V2 5 V = Equation Editor -5 1 T(v) 4v, T(v): = Levi defines a very particular linear map T: R R which stretches the vector V 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 (-8v +6v2 +7v3). V3 = = -3 .:). -8 1 1 -V, T(3)= - V3. 3 A- A I B I US X X 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 Equation Editor (d) Help Levi find A. A = Equation G Q Editor 12 X == A- A I B IU S x x Styles Font Size (e) Having found A, Levi immediately writes down A without performing any row operations or matrix multiplication. Explain why Levi was able to do this. 20 X DX GGG Q 13 *= S Words: 0 Font A- A I B I US X X Styles Size Y & A Words: 0
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