A-LevelFurther MathematicsMatrices9231/12 · May/June 2014 · Q68 Marks
The linear transformation T : R⁴ → R⁴ is represented by the matrix M, where M = [ [2, -1, 1, 3], [2, 0, 0, 5], [6, -2, 2, 11], [10, -3, 3, 19] ]. (i) Find the rank of M and state a basis for the range space of T. [4] (ii) Obtain a basis for the null space of T. [4]
- (i)Find the rank of M and state a basis for the range space of T.[4]
- (ii)Obtain a basis for the null space of T.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2014). The mark or mark type is shown in brackets.
- [M1A1] Row reduction of M to [ [2, -1, 1, 3], [0, 1, -1, 2], [0, 0, 0, 0], [0, 0, 0, 0] ]
- [A1] R(M) = 4-2 = 2
- [A1] Basis for range space consists of any two independent column vectors from matrix M. e.g. (2, 2, 6, 10)^T, (-1, 0, -2, -3)^T
- [M1] Form system of equations from RREF of M: 2x - y + z + 3t = 0, y - z + 2t = 0
- [M1] t = λ, z = μ, y = μ - 2λ, x = -5/2 λ (OE)
- [A1A1] Basis for null space is ( -5/2, -2, 0, 1 )^T, ( 0, 1, 1, 0 )^T (OE)
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