(a) Suppose that the vectors {v1, v2, · · · , Vn} is a finite set of linearly independents vectors in some infinite dimensional vector space V. Prove that {v1, V2, · · · , Vn} do NOT span V. (Hint: Use a proof by contradiction) ' ... pose T is a linear transformation, T : P11(R) → M24(R) Sup Further suppose that dim(ker(T)) = 4 Prove that T will be onto. (c) Consider the mapping T: P2(R) → P4(R) T(p(x)) = x²p(x) Prove that T is a linear transformation.

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.