Page 655 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(d) ,

Show that the inverse of a bijective transformation from V to W is a bijective transformation from W to V. Also, show that the
6. inverse of a bijective linear transformation is a bijective linear transformation.

Prove: There can be a surjective linear transformation from V to W only if .
7.

8. symmetric matrices and .
(a) Find an isomorphism between the vector space of all

(b) Find two different isomorphisms between the vector space of all matrices and .

(c) Find an isomorphism between the vector space of all polynomials of degree at most 3 such that and .

(d) Find an isomorphism between the vector space and .

Let S be the standard basis for . Prove Theorem 8.6.2 by showing that the linear transformation that maps
9. to its coordinate vector in is an isomorphism.

Show that if , are bijective linear transformations, then the composition is a bijective linear transformation.
10.

11. (For Readers Who Have Studied Calculus) be computed by matrix
.
How could differentiation of functions in the vector space
multiplication in ? Use your method to find the derivative of

Isomorphisms preserve the algebraic structure of vector spaces. The geometric structure depends

12. on notions of angle and distance and so, ultimately, on the inner product. If V and W are

finite-dimensional inner product spaces, then we say that is an inner product space

isomorphism if it is an isomorphism between V and W, and furthermore,

That is, the inner product of u and v in V is equal to the inner product of their images in W.

(a) Prove that an inner product space isomorphism preserves angles and distances—that is, the
angle between u and v in V is equal to the angle between and in W, and
.