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3.For the vector space of all polynomials int of degree at most 3 and the scalar product on this space given by (p(t),q(t

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MA 1112: Linear Algebra II Tutorial problems, March 12, 2019

1.For the space R3 with the standard scalar product, find the orthogonal basis e1, e2, e3 obtained by Gram–Schmidt orthogonalisation from f1 =

 1 3 3

,

f2=

 1 2 4

, f3=

 1 1 1

.

2.Show that the formula (

µx1

y1

¶ ,

µx2

y2

)=x1x2+1

2(x1y2+x2y1)+y1y2.

defines a scalar product onR2, and find an orthonormal basis ofR2with respect to that scalar product.

3.For the vector space of all polynomials int of degree at most 3 and the scalar product on this space given by

(p(t),q(t))= Z 1

−1

p(t)q(t)d t,

find the result of Gram–Schmidt orthogonalisation of the vectors 1,t,t2,t3. Optional question: Show that inRn, it is impossible to findn+2 vectors that only form obtuse angles (that is, (vi,vj)<0 for alli 6=j).

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