A-LevelFurther MathematicsVector SpacesMay/June 2013Paper 1 Q810 Marks
The linear transformations T₁ : R⁴ → R⁴ and T₂ : R⁴ → R⁴ are represented by the matrices M₁ and M₂ respectively, where M₁ = [[1, -2, 3, 5], [3, -4, 17, 33], [5, -9, 20, 36], [4, -7, 16, 29]] and M₂ = [[1, -2, 0, -3], [2, -1, 0, 0], [4, -7, 1, -9], [6, -10, 0, -14]]. The null spaces of T₁ and T₂ are denoted by K₁ and K₂ respectively. Find a basis for K₁ and a basis for K₂. It is given that a = [[1], [2], [3], [4]]. The vectors x₁ and x₂ are such that M₁x₁ = M₁a and M₂x₂ = M₂a. Given that x₁ - x₂ = [[p], [5], [7], [q]], find p and q.
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