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(b) Compute the matrix Aϕ,v of the linear transformation ϕ relative to the basis of standard unit vectors v1 = 1 0 , v2 = 0 1

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MA 1111: Linear Algebra I Tutorial problems, November 22, 2018

1. The set of all complex numbers forms a 2-dimensional vector space over real num- bers with a basis 1, i. Compute, relative to this basis, the matrix of the linear operator on that space which maps every complex number z to(3−7i)z.

2. LetV =R2, e1 = 1

1

, e2 = 2

3

a basis of V, ϕ: V →V a linear transformation whose matrix Aϕ,e relative to the basis e1, e2 is

0 1 1 0

.

(a) Find the transition matrix Mef from the basis e1, e2 to the basis f1 = 1

−2

, f2=

4

−9

, and compute the matrixAϕ,f.

(b) Compute the matrix Aϕ,v of the linear transformation ϕ relative to the basis of standard unit vectors v1 =

1 0

, v2 =

0 1

.

3. Does there exist a change of basis making the matrix A=

3 0 −1 5 −1 −8

−1 1 4

 diag- onal? Why?

4. For the matrix A=

3 2 1 2

, find a closed formula for An.

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