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Chapter 4: Q 4.3-69E (page 198)

Question: Consider a basis f1,....,fn of Pn-1 .Let a1,....,an be distinct real numbers. Consider the n 脳 n matrix M whose ijth entry is fj ( ai ) .Show that the matrix m is invertible.

Short Answer

Expert verified

The matrix m is invertible

Step by step solution

01

Definition of isomorphism

An invertible linear transformation T is called an isomorphism.

Also says that the linear space V is isomorphic to the linear space W if there exists an isomorphism T from V to W.

02

Step:2 Determination of the invertible matrix

Consider a basis f1,...,fn of Pn-1.

Let a1,...,an be distinct real numbers.

Consider the n脳n matrix M whose ijth entry is fjai.

Now to show that the matrix m is invertible.

Assume that the basis B=f1,...,fn is the basis of a linear space V then the coordinate transformation LBf=fB from V to Rn is an isomorphism.

Thus from the above assumption the given matrix m is invertible.

Hence the proof.

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