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Chapter 5: Q2E (page 260)

Does the equation

<f,g+h>=<f,g>+<f,h>

Hold for all elements f, g, hof an inner product space? Explain.

Short Answer

Expert verified

<f,g+h>=<f,g>+<f,h>

Step by step solution

01

Determine the symmetric property.

Here it needs to prove that for any f,g,h鈭�V,Vbeing an inner product space,

f,g+h=f,g+f,h.

Thus, consider the solution below with the help of symmetric property.

f,g+h=g+h,f=g,f+f,h=f,g+f,h

Hence, it is proved that f,g+h=f,g+f,h.

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Most popular questions from this chapter

a.Find all n脳nmatrices that are both orthogonal and upper triangular, with positive diagonal entries.

b.Show that the QRfactorization of an invertible n脳nmatrix is unique. Hint: If,A=Q1R1=Q2R2 thenthe matrixA=Q2-1Q1=Q2R1-1 is both orthogonal and upper triangular, with positive diagonal entries.

Find the angle胃between each of the pairs of vectors u鈫�and localid="1659433601917" v鈫�in exercises 4 through

6.u鈫�=[1-12-2,v鈫�=2345]

If A and B are arbitrary n脳nmatrices, which of the matrices in Exercise 21 through 26 must be symmetric?

BBT.

Is there an orthogonal transformation T from R3to R3such thatT[230]=[302]

and T[230]=[302]?

If A and B are arbitraryn脳n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?

B(A+A螕)B螕.
See all solutions

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