Page 401 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let be a basis for a vector space V. Show that is also a basis, where , , and
23. .

24. linearly independent vectors in .
(a) Show that for every positive integer n, one can find

Hint Look for polynomials.

(b) Use the result in part (a) to prove that is infinite-dimensional.

(c) Prove that , , and are infinite-dimensional vector spaces.

Let S be a basis for an n-dimensional vector space V. Show that if form a linearly independent set of vectors in V,

25. then the coordinate vectors form a linearly independent set in , and conversely.

Using the notation from Exercise 25, show that if span V, then the coordinate vectors span
26. , and conversely.

Find a basis for the subspace of spanned by the given vectors.
27.

(a)

(b)

(c)

Hint Let S be the standard basis for and work with the coordinate vectors relative to S; note Exercises 25 and 26.

The accompanying figure shows a rectangular -coordinate system and an -coordinate system with skewed axes.
28. Assuming that 1-unit scales are used on all the axes, find the -coordinates of the points whose -coordinates are given.

(a) (1, 1)

(b) (1, 0)

(c) (0, 1)

(d) (a, b)