Intermediate Algebra
10th Edition
ISBN: 9781305191495
Author: Jerome E. Kaufmann; Karen L. Schwitters
Publisher: Cengage Learning US
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Chapter 8.CR, Problem 34CR
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Let
2
A =
4
3
-4
0
1
(a) Show that v =
eigenvalue.
()
is an eigenvector of A and find the corresponding
(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)
may be useful.
(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.
(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.
(c) Let
6
0 0
A =
-10 4 8
5 1 2
(i) Find the characteristic polynomial of A and factorise it.
(ii) Determine all eigenvalues of A and find bases for the corresponding
eigenspaces.
(iii) Is A diagonalisable? Give reasons for your answer.
most 2, and let
Let P2 denote the vector space of polynomials of degree at
D: P2➡ P2
be the transformation that sends a polynomial p(t) = at² + bt+c in P2 to its derivative
p'(t)
2at+b, that is,
D(p) = p'.
(a) Prove that D is a linear transformation.
(b) Find a basis for the kernel ker(D) of the linear transformation D and compute its
nullity.
(c) Find a basis for the image im(D) of the linear transformation D and compute its
rank.
(d) Verify that the Rank-Nullity Theorem holds for the linear transformation D.
(e) Find the matrix representation of D in the standard basis (1,t, t2) of P2.
Chapter 8 Solutions
Intermediate Algebra
Ch. 8.1 - For Problems 110, answer true or false. The graph...Ch. 8.1 - Prob. 2CQCh. 8.1 - Prob. 3CQCh. 8.1 - Prob. 4CQCh. 8.1 - Prob. 5CQCh. 8.1 - Prob. 6CQCh. 8.1 - Prob. 7CQCh. 8.1 - Prob. 8CQCh. 8.1 - Prob. 9CQCh. 8.1 - Prob. 10CQ
Ch. 8.1 - Prob. 1PSCh. 8.1 - Prob. 2PSCh. 8.1 - Prob. 3PSCh. 8.1 - Prob. 4PSCh. 8.1 - Prob. 5PSCh. 8.1 - Prob. 6PSCh. 8.1 - Prob. 7PSCh. 8.1 - Prob. 8PSCh. 8.1 - Prob. 9PSCh. 8.1 - Prob. 10PSCh. 8.1 - Prob. 11PSCh. 8.1 - Prob. 12PSCh. 8.1 - Prob. 13PSCh. 8.1 - Prob. 14PSCh. 8.1 - Prob. 15PSCh. 8.1 - Prob. 16PSCh. 8.1 - Prob. 17PSCh. 8.1 - Prob. 18PSCh. 8.1 - Prob. 19PSCh. 8.1 - Prob. 20PSCh. 8.1 - Prob. 21PSCh. 8.1 - Prob. 22PSCh. 8.1 - Prob. 23PSCh. 8.1 - Prob. 24PSCh. 8.1 - Prob. 25PSCh. 8.1 - Prob. 26PSCh. 8.1 - Prob. 27PSCh. 8.1 - Prob. 28PSCh. 8.1 - Prob. 29PSCh. 8.1 - Prob. 30PSCh. 8.1 - Prob. 31PSCh. 8.1 - Prob. 32PSCh. 8.1 - Prob. 33PSCh. 8.1 - Prob. 34PSCh. 8.1 - Prob. 35PSCh. 8.1 - Prob. 36PSCh. 8.1 - Prob. 37PSCh. 8.1 - Prob. 38PSCh. 8.1 - Prob. 39PSCh. 8.2 - Prob. 1CQCh. 8.2 - Prob. 2CQCh. 8.2 - Prob. 3CQCh. 8.2 - Prob. 4CQCh. 8.2 - Prob. 5CQCh. 8.2 - Prob. 6CQCh. 8.2 - Prob. 7CQCh. 8.2 - Prob. 8CQCh. 8.2 - Prob. 9CQCh. 8.2 - Prob. 10CQCh. 8.2 - Prob. 1PSCh. 8.2 - Prob. 2PSCh. 8.2 - Prob. 3PSCh. 8.2 - Prob. 4PSCh. 8.2 - Prob. 5PSCh. 8.2 - Prob. 6PSCh. 8.2 - Prob. 7PSCh. 8.2 - Prob. 8PSCh. 8.2 - Prob. 9PSCh. 8.2 - Prob. 10PSCh. 8.2 - Prob. 11PSCh. 8.2 - Prob. 12PSCh. 8.2 - Prob. 13PSCh. 8.2 - Prob. 14PSCh. 8.2 - Prob. 15PSCh. 8.2 - Prob. 16PSCh. 8.2 - Prob. 17PSCh. 8.2 - Prob. 18PSCh. 8.2 - Prob. 19PSCh. 8.2 - Prob. 20PSCh. 8.2 - Prob. 21PSCh. 8.2 - Prob. 22PSCh. 8.2 - Prob. 23PSCh. 8.2 - Prob. 24PSCh. 8.2 - Prob. 25PSCh. 8.2 - Prob. 26PSCh. 8.2 - Prob. 27PSCh. 8.2 - Prob. 28PSCh. 8.2 - Prob. 29PSCh. 8.2 - Prob. 30PSCh. 8.2 - Prob. 31PSCh. 8.2 - Prob. 32PSCh. 8.2 - Prob. 33PSCh. 8.2 - Prob. 34PSCh. 8.2 - Prob. 35PSCh. 8.2 - Prob. 36PSCh. 8.2 - Prob. 37PSCh. 8.2 - Prob. 38PSCh. 8.2 - Prob. 39PSCh. 8.2 - Prob. 40PSCh. 8.2 - Prob. 41PSCh. 8.2 - Prob. 42PSCh. 8.2 - Prob. 43PSCh. 8.2 - Prob. 44PSCh. 8.2 - Prob. 45PSCh. 8.2 - Prob. 46PSCh. 8.2 - Prob. 47PSCh. 8.2 - Prob. 48PSCh. 8.2 - Prob. 49PSCh. 8.2 - Prob. 50PSCh. 8.2 - Prob. 51PSCh. 8.2 - Prob. 52PSCh. 8.2 - Prob. 53PSCh. 8.2 - Prob. 54PSCh. 8.2 - Prob. 55PSCh. 8.2 - Prob. 56PSCh. 8.2 - Prob. 57PSCh. 8.2 - Prob. 58PSCh. 8.2 - Prob. 59PSCh. 8.2 - Prob. 60PSCh. 8.2 - Prob. 61PSCh. 8.2 - Prob. 62PSCh. 8.2 - Prob. 63.1PSCh. 8.2 - By expanding (xh)2+(yk)2=r2, we obtain...Ch. 8.2 - Prob. 63.3PSCh. 8.2 - Prob. 63.4PSCh. 8.2 - Prob. 63.5PSCh. 8.2 - Prob. 63.6PSCh. 8.2 - Prob. 64PSCh. 8.2 - Prob. 65PSCh. 8.2 - Prob. 66.1PSCh. 8.2 - Prob. 66.2PSCh. 8.2 - Prob. 66.3PSCh. 8.2 - Prob. 66.4PSCh. 8.2 - Prob. 66.5PSCh. 8.2 - Prob. 66.6PSCh. 8.3 - Prob. 1CQCh. 8.3 - Prob. 2CQCh. 8.3 - Prob. 3CQCh. 8.3 - Prob. 4CQCh. 8.3 - Prob. 5CQCh. 8.3 - Prob. 6CQCh. 8.3 - Prob. 7CQCh. 8.3 - Prob. 8CQCh. 8.3 - Prob. 9CQCh. 8.3 - Prob. 10CQCh. 8.3 - Prob. 1PSCh. 8.3 - Prob. 2PSCh. 8.3 - Prob. 3PSCh. 8.3 - Prob. 4PSCh. 8.3 - Prob. 5PSCh. 8.3 - Prob. 6PSCh. 8.3 - Prob. 7PSCh. 8.3 - Prob. 8PSCh. 8.3 - Prob. 9PSCh. 8.3 - Prob. 10PSCh. 8.3 - Prob. 11PSCh. 8.3 - Prob. 12PSCh. 8.3 - Prob. 13PSCh. 8.3 - Prob. 14PSCh. 8.3 - Prob. 15PSCh. 8.3 - Prob. 16PSCh. 8.3 - Prob. 17PSCh. 8.3 - Prob. 18PSCh. 8.3 - Prob. 19PSCh. 8.3 - Prob. 20PSCh. 8.3 - Prob. 21PSCh. 8.3 - Prob. 22PSCh. 8.3 - Prob. 23PSCh. 8.3 - Prob. 24PSCh. 8.3 - Prob. 25PSCh. 8.3 - Prob. 26PSCh. 8.3 - Prob. 27PSCh. 8.3 - Prob. 28PSCh. 8.3 - Prob. 29PSCh. 8.3 - Prob. 30PSCh. 8.4 - Prob. 1CQCh. 8.4 - Prob. 2CQCh. 8.4 - Prob. 3CQCh. 8.4 - Prob. 4CQCh. 8.4 - Prob. 5CQCh. 8.4 - Prob. 6CQCh. 8.4 - Prob. 7CQCh. 8.4 - Prob. 8CQCh. 8.4 - Prob. 9CQCh. 8.4 - Prob. 10CQCh. 8.4 - Prob. 1PSCh. 8.4 - Prob. 2PSCh. 8.4 - Prob. 3PSCh. 8.4 - Prob. 4PSCh. 8.4 - Prob. 5PSCh. 8.4 - Prob. 6PSCh. 8.4 - Prob. 7PSCh. 8.4 - Prob. 8PSCh. 8.4 - Prob. 9PSCh. 8.4 - Prob. 10PSCh. 8.4 - Prob. 11PSCh. 8.4 - Prob. 12PSCh. 8.4 - Prob. 13PSCh. 8.4 - Prob. 14PSCh. 8.4 - Prob. 15PSCh. 8.4 - Prob. 16PSCh. 8.4 - Prob. 17PSCh. 8.4 - Prob. 18PSCh. 8.4 - Prob. 19PSCh. 8.4 - Prob. 20PSCh. 8.4 - Prob. 21PSCh. 8.4 - Prob. 22PSCh. 8.4 - Prob. 23PSCh. 8.4 - Prob. 24PSCh. 8.4 - Prob. 25PSCh. 8.4 - Prob. 26PSCh. 8.4 - Prob. 27PSCh. 8.4 - Prob. 28PSCh. 8.4 - Prob. 29PSCh. 8.4 - Prob. 30PSCh. 8.4 - Prob. 31PSCh. 8.4 - Prob. 32PSCh. 8.4 - Prob. 33PSCh. 8.4 - Prob. 34PSCh. 8.4 - Prob. 35PSCh. 8.4 - Prob. 36PSCh. 8.4 - Prob. 37PSCh. 8.4 - Prob. 38PSCh. 8.4 - Prob. 39PSCh. 8.4 - Prob. 40.1PSCh. 8.4 - Prob. 40.2PSCh. 8.4 - Prob. 40.3PSCh. 8.4 - Prob. 40.4PSCh. 8.4 - Prob. 40.5PSCh. 8.4 - Prob. 40.6PSCh. 8.4 - Prob. 41.1PSCh. 8.4 - Prob. 41.2PSCh. 8.4 - Prob. 41.3PSCh. 8.4 - Prob. 41.4PSCh. 8.4 - Prob. 41.5PSCh. 8.4 - Prob. 41.6PSCh. 8.4 - Prob. 41.7PSCh. 8.4 - Prob. 41.8PSCh. 8.4 - Prob. 41.9PSCh. 8.4 - Prob. 41.10PSCh. 8.4 - Prob. 42PSCh. 8.S - Prob. 1SCh. 8.S - Prob. 2SCh. 8.S - Prob. 3SCh. 8.S - Prob. 4SCh. 8.S - Prob. 5SCh. 8.S - Prob. 6SCh. 8.S - Prob. 7SCh. 8.S - Prob. 8SCh. 8.CR - Prob. 1CRCh. 8.CR - Prob. 2CRCh. 8.CR - Prob. 3CRCh. 8.CR - Prob. 4CRCh. 8.CR - Prob. 5CRCh. 8.CR - Prob. 6CRCh. 8.CR - Prob. 7CRCh. 8.CR - Prob. 8CRCh. 8.CR - Prob. 9CRCh. 8.CR - Prob. 10CRCh. 8.CR - Prob. 11CRCh. 8.CR - Prob. 12CRCh. 8.CR - Prob. 13CRCh. 8.CR - Prob. 14CRCh. 8.CR - Prob. 15CRCh. 8.CR - Prob. 16CRCh. 8.CR - Prob. 17CRCh. 8.CR - Prob. 18CRCh. 8.CR - Prob. 19CRCh. 8.CR - Prob. 20CRCh. 8.CR - Prob. 21CRCh. 8.CR - Prob. 22CRCh. 8.CR - Prob. 23CRCh. 8.CR - Prob. 24CRCh. 8.CR - Prob. 25CRCh. 8.CR - Prob. 26CRCh. 8.CR - Prob. 27CRCh. 8.CR - Prob. 28CRCh. 8.CR - Prob. 29CRCh. 8.CR - Prob. 30CRCh. 8.CR - Prob. 31CRCh. 8.CR - Prob. 32CRCh. 8.CR - Prob. 33CRCh. 8.CR - For Problems 3150, graph each equation....Ch. 8.CR - Prob. 35CRCh. 8.CR - Prob. 36CRCh. 8.CR - Prob. 37CRCh. 8.CR - Prob. 38CRCh. 8.CR - Prob. 39CRCh. 8.CR - Prob. 40CRCh. 8.CR - Prob. 41CRCh. 8.CR - Prob. 42CRCh. 8.CR - Prob. 43CRCh. 8.CR - Prob. 44CRCh. 8.CR - Prob. 45CRCh. 8.CR - Prob. 46CRCh. 8.CR - Prob. 47CRCh. 8.CR - Prob. 48CRCh. 8.CR - Prob. 49CRCh. 8.CR - Prob. 50CRCh. 8.CT - Prob. 1CTCh. 8.CT - Prob. 2CTCh. 8.CT - Prob. 3CTCh. 8.CT - Prob. 4CTCh. 8.CT - Prob. 5CTCh. 8.CT - Prob. 6CTCh. 8.CT - Prob. 7CTCh. 8.CT - Prob. 12CTCh. 8.CT - Prob. 13CTCh. 8.CT - Prob. 14CTCh. 8.CT - Prob. 15CTCh. 8.CT - Prob. 16CTCh. 8.CT - Prob. 17CTCh. 8.CT - Prob. 18CTCh. 8.CT - Prob. 19CTCh. 8.CT - Prob. 20CTCh. 8.CT - Prob. 21CTCh. 8.CT - Prob. 22CTCh. 8.CT - Prob. 23CTCh. 8.CT - Prob. 24CTCh. 8.CT - Prob. 25CTCh. 8.CM - Prob. 1CMCh. 8.CM - Prob. 2CMCh. 8.CM - Prob. 3CMCh. 8.CM - Prob. 4CMCh. 8.CM - Prob. 5CMCh. 8.CM - Prob. 6CMCh. 8.CM - Prob. 7CMCh. 8.CM - Prob. 8CMCh. 8.CM - Prob. 9CMCh. 8.CM - Prob. 10CMCh. 8.CM - Prob. 11CMCh. 8.CM - Prob. 12CMCh. 8.CM - Prob. 13CMCh. 8.CM - Prob. 14CMCh. 8.CM - Prob. 15CMCh. 8.CM - Prob. 16CMCh. 8.CM - Prob. 17CMCh. 8.CM - Prob. 18CMCh. 8.CM - Prob. 19CMCh. 8.CM - Prob. 20CMCh. 8.CM - Prob. 21CMCh. 8.CM - Prob. 22CMCh. 8.CM - Prob. 23CMCh. 8.CM - Prob. 24CMCh. 8.CM - Prob. 25CMCh. 8.CM - Prob. 26CMCh. 8.CM - Prob. 27CMCh. 8.CM - Prob. 28CMCh. 8.CM - Prob. 29CMCh. 8.CM - Prob. 30CMCh. 8.CM - Prob. 31CMCh. 8.CM - Prob. 32CMCh. 8.CM - Prob. 33CMCh. 8.CM - Prob. 34CMCh. 8.CM - Prob. 35CMCh. 8.CM - Prob. 36CMCh. 8.CM - Prob. 37CMCh. 8.CM - Prob. 38CMCh. 8.CM - Prob. 39CMCh. 8.CM - Prob. 40CMCh. 8.CM - Prob. 41CMCh. 8.CM - Prob. 42CMCh. 8.CM - Prob. 43CMCh. 8.CM - Prob. 44CMCh. 8.CM - Prob. 45CMCh. 8.CM - Prob. 46CMCh. 8.CM - Prob. 47CMCh. 8.CM - Prob. 48CMCh. 8.CM - Prob. 49CMCh. 8.CM - Prob. 50CMCh. 8.CM - Prob. 51CMCh. 8.CM - Prob. 52CMCh. 8.CM - Prob. 53CMCh. 8.CM - Prob. 54CMCh. 8.CM - Prob. 55CMCh. 8.CM - Prob. 56CMCh. 8.CM - For Problems 5564, solve inequality and express...Ch. 8.CM - Prob. 58CMCh. 8.CM - Prob. 59CMCh. 8.CM - Prob. 60CMCh. 8.CM - Prob. 61CMCh. 8.CM - Prob. 62CMCh. 8.CM - Prob. 63CMCh. 8.CM - Prob. 64CMCh. 8.CM - Prob. 65CMCh. 8.CM - For Problems 65-70, graph the following equations....Ch. 8.CM - Prob. 67CMCh. 8.CM - Prob. 68CMCh. 8.CM - Prob. 69CMCh. 8.CM - Prob. 70CMCh. 8.CM - Prob. 71CMCh. 8.CM - Prob. 72CMCh. 8.CM - Prob. 73CMCh. 8.CM - Prob. 74CMCh. 8.CM - Prob. 75CMCh. 8.CM - Prob. 76CMCh. 8.CM - Prob. 77CMCh. 8.CM - Prob. 78CMCh. 8.CM - Prob. 79CMCh. 8.CM - Prob. 80CMCh. 8.CM - Prob. 81CMCh. 8.CM - Prob. 82CMCh. 8.CM - Prob. 83CMCh. 8.CM - Prob. 84CMCh. 8.CM - Prob. 85CMCh. 8.CM - Prob. 86CMCh. 8.CM - Prob. 87CMCh. 8.CM - Prob. 88CM
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