A-LevelFurther MathematicsLinear Spaces and TransformationsMay/June 2012Paper 1 Q710 Marks
The linear transformations T₁ : R⁴ → R⁴ and T₂ : R⁴ → R⁴ are represented by the matrices M₁ = [[1, 1, 1, 4], [2, 1, 4, 11], [3, 4, 1, 9], [4, -3, 18, 37]] and M₂ = [[1, 1, 1, -1], [2, 3, 0, 1], [3, 4, 1, 0], [4, 5, 2, 0]] respectively. The null space of T₁ is denoted by K₁ and the null space of T₂ is denoted by K₂. Show that the dimension of K₁ is 2 and that the dimension of K₂ is 1. Find the basis of K₁ which has the form [[p], [q], [r], [s]] such that [[r], [s]] = [[1], [0]] and show that K₂ is a subspace of K₁.
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