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1.(20 marks) The subspaceU inR3 is the linear span of the vectors

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MA1112: MIDTERM TEST

FEBRUARY25, 2019 Name:

Student number:

Course of study (Maths / TP / TSM / Visiting student):

Instructions: Attempt all questions. Please attach this sheet to the solutions you hand in. Write your name and student number legibly on each sheet you are handing in. You may use without proof all statements proved in class, or stated in homework / tutorial questions.

1.(20 marks) The subspaceU inR3 is the linear span of the vectors

−1 0 1

and

 3

−2

−1

, and the subspaceV is the solution set to the equationx+y+4z=0.

Describe the intersectionU∩V. Find a basis ofU relative toU∩V, and a basis ofV relative toU∩V.

2.(20 marks) Is the 2D plane inR3defined by the equationx−2y−z=0 an invariant subspace of the linear transformationϕthat multiplies every vector by the matrix

−26 20 21 11 −9 −9

−49 40 40

? Explain your answer.

3.(35 marks) Consider the linear transformationϕofR3that multiplies ev- ery vector by the matrix

6 −23 14 3 −16 10 2 −14 9

. Determine the Jordan form of this linear

transformation, and find some Jordan basis for it.

4.(25 marks) Given that for a linear transformation ϕ: R4→ R4 we have rk(ϕ2)=2, what are possible values of null(ϕ)? Explain your answer, and for each value you predict, give an example of a transformation ϕfor which that value is correct.

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