Page 418 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

For the matrices in Exercise 6, find a basis for the column space of A.
9.

For the matrices in Exercise 6, find a basis for the row space of A consisting entirely of row vectors of A.
10.

Find a basis for the subspace of spanned by the given vectors.
11.

(a) (1, 1, −4, −3), (2, 0, 2, −2), (2, −1, 3, 2)

(b) (−1, 1, −2, 0), (3, 3, 6, 0), (9, 0, 0, 3)

(c) (1, 1, 0, 0), (0, 0, 1, 1), (−2, 0, 2, 2), (0, −3, 0, 3)

Find a subset of the vectors that forms a basis for the space spanned by the vectors; then express each vector that is not in the
12. basis as a linear combination of the basis vectors.

(a) , , , ,
(b) , , ,
(c) , , ,

Prove that the row vectors of an invertible matrix A form a basis for .
13.

14.
(a) Let

and consider a rectangular -coordinate system in 3-space. Show that the nullspace of A
consists of all points on the z-axis and that the column space consists of all points in the
-plane (see the accompanying figure).

(b) Find a matrix whose nullspace is the x-axis and whose column space is the -plane.