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

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MA 1212: Linear Algebra II Tutorial problems, January 29, 2015

1.For the spaceR3 with the standard inner product, find the orthogonal basise1,e2,e3obtained by Gram–Schmidt orthogonalisation fromf1=

 1 3 3

,

f2 =

 1 2 4

, f3=

 1 1 1

.

2.Show that the formula (

x1 y1

,

x2 y2

) =x1x2+1

2(x1y2+x2y1) +y1y2.

defines a scalar product on R2, and find an orthonormal basis of R2 with respect to that scalar product.

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

(p(t), q(t)) = Z1

−1

p(t)q(t)dt,

find the result of Gram–Schmidt orthogonalisation of the vectors 1, t, t2, t3.

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