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(a) For the vector spaceR3, show that the vectors e1

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MA 1111: Linear Algebra I Tutorial problems, November 15, 2018

1. (a) For the vector spaceR3, show that the vectors

e1 =

 0 2

−1

, e2 =

 1 1 1

, e3 =

 2 1 3

form a basis, and so do

f1 =

 1 1 1

, f2 =

−1

−1 2

, f3 =

 1

−2

−1

.

(b) For the vector space R3, find the transition matrix Me,f for the two bases from the previous question.

(c) Given that a vector has coordinates1, 4,−3with respect to the basisf1, f2, f3, find its coordinates with respect to the basis e1, e2, e3.

(d) Given that a vector has coordinates 1, 4,−3 with respect to the basis e1, e2, e3, find its coordinates with respect to the basis f1, f2, f3.

In the following two questions, we denote by Pn the vector space of all polynomials inx of degree at most n.

2.Write down the transition matrix between the bases 1,x, . . . , xn and 1,x+1, . . . , (x+1)n of Pn for (a)n=1; (b)n=2; (c) n=3.

3. Which of the following functions from P3 toP3 are linear operators? Explain your answers. When a function is a linear operator, write down its matrix relative to the standard basis1, x, x2, x3.

(a) f(x)7→ f(x)−f(0)x ; (b)f(x)7→xf0(x) −2f(x); (c) f(x)7→f00(x)f0(x) −t2f000(x).

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