A-LevelFurther MathematicsMatrices9231/12 · May/June 2018 · Q810 Marks
The linear transformation T : R⁴ → R³ is represented by the matrix M, where M = ⎛ 1 2 α -1 ⎞ ⎜ 2 6 -3 -3 ⎟ ⎝ 3 10 -6 -5 ⎠ and α is a constant. When α ≠ 0 the null space of T is denoted by K₁.
- (i)Find a basis for K₁.[5]
- (ii)When α = 0 the null space of T is denoted by K₂. Find a basis for K₂.[3]
- (iii)Determine, justifying your answer, whether K₁ is a subspace of K₂.[2]
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- M1 (i) Good attempt at REF. Operations R2-2R1, R3-3R1
- A1 (i) (1 2 α -1 / 0 2 -3-2α -1 / 0 0 α 0)
- M1 (i) Forms system of equations from row echelon matrix. x+2y+αz-t=0, 2y-(3+2α)z-t=0, αz=0
- A1 (i) Three correct equations. Since α≠0, z=0.
- A1 (i) (K₁ =) (0 1 0 2)^T
- M1 (ii) If α=0 then forms system of 2 equations: x+2y-t=0 and 2y-3z-t=0
- A1 (ii) Basis vector 1: (-3, 0, 1, -3)^T (AEF)
- A1 (ii) Basis vector 2: (0, 1, 0, 2)^T (AEF)
- M1 (iii) Credit should be given for a correct conclusion using their bases.
- A1 FT (iii) Therefore K₁ is a subspace of K₂.
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