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§3.5

Hoffman 1' Kunze Solutions

p. 10ó ^ 2

Lot B = {oi,q₂,q₃} 1>c a basis for C³ dcfined by: «i = (1.0. -1), a₂ = (1,1,1). and o₃ = (2.2.0). Flnd thc dual basis of B.

Lot B* = {if\. </.«• </<} be the dual basis for B and {/j./₂,/₃} bo the dual basis for tho standard basis. Them g\ = a\J\ + 6j/₂ + cj/₃. Hence we havo tho following system of equatious:

Tho matrix for tliis system is:

(hM =	01 - Cl	= 1
<jiM =	fl| +61 +Ci	= 0
9i(<*s) =	2a, +25,	= 0
0 -i| i>	i /	1 0	0	1 \
1 10		0 1	0	-^1 .
1 0 1 0,	^1 \	0 0	1	0/

A =

Now wo havo g\ - J\ ~ Si-

A similar solution shows g₂ =    - /₂ + /, and </, = -3/1 + fi - 5/3.

P. 105 2:3

If .4 and fi aro n x n matrices ovor tho fiold F. show that traco(.4/?) = traco(/?.4). Now show that similar matricos have the same tracę.

Lot C = AB - (c,>) Tlicn r,j =    So c„ = a,kbn. Now:

n n    n n

traco(.4£?) =    ~Y,Y,^bk'^a'^k = tracc(/i.4).

i-l fc = l    fc=l i=l

Now if .4 is similar to B thcn 3 P such that B = P ^lAP. traoo(P) = tracc(P ¹ -4/^>) = traco(.4P ¹P) — traco(.4).

p. 106 - 4

Lot \* l»c the voctor space of all polynomial functions froin R to R which have <logroo 2 or less: p(x) = c₀ + c,x + cjt². Defino throo linear functionals on V by

f\(p) = ( p(r)d-r, fi(p)= f pix)dr. fsip) = f p(x)<lx.

Jo    Jo    Jo

Show that {/1./2 /3} i^{s a} basis for V* by oxibiting a basis for V of which it is the dual.

Wo wish to hnd a basis B = {/>i./>,./>;$} C V such that {/i./a,/₃} = B‘.

{PM = cio + c_ux + c_l2x-p₂(i) = C2o + ^c2i* + c₂₂x² then we havc:

P.M = c-m+ c_3lx + c₃₂x²

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