Page 603 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) What is the dimension of the solution space of ?

(b) Is consistent for all vectors in ? Explain.

Let be a linear transformation from to any vector space. Show that the kernel of T is a line through the
19. origin, a plane through the origin, the origin only, or all of .

Let be a linear transformation from any vector space to . Show that the range of T is a line through the
20. origin, a plane through the origin, the origin only, or all of .

Let be multiplication by
21.

(a) Show that the kernel of T is a line through the origin, and find parametric equations for it.
(b) Show that the range of T is a plane through the origin, and find an equation for it.

Prove: If is a basis for V and , , …, are vectors in W, not necessarily distinct, then there exists a
22. linear transformation such that

For the positive integer , let be the linear transformation defined by , for A an matrix

23. with real entries. Determine the dimension of .

Prove the dimension theorem in the cases
24.

(a)

(b)

25. (For Readers Who Have Studied Calculus) Let be the differentiation transformation .
Describe the kernel of D. be the integration transformation .

26. (For Readers Who Have Studied Calculus) Let
Describe the kernel of J.