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**preview**shows half of the first page. to view the full**2 pages of the document.**MAT 247S - Problem Set 1

Due Thursday January 22nd

NOTE: Questions 1 d), 1 e), 2 a), 5 and 7 c) will be marked.

1. Determine whether the indicated function hĀ·,Ā·i deļ¬nes an inner product on the vector space

V. If it is an inner product, prove that each of the four conditions in the deļ¬nition of inner

product is satisļ¬ed. If it is not an inner product, demonstrate (with an explicit example) how

one of the conditions fails to hold.

a) Let Vbe the vector space of continuous functions from the interval [0,1] to the real

numbers R. Set

hf1, f2i=īZ1

0

f1(t)f2(t)dtī+f1(0)f2(0), f1, f2āV.

b) Let V=C2and A=ī5ā2i

āā2i3ī. Set hx, yi=xAyā,x,yāC2. (Here, if y= (y1, y2),

then yā=īĀÆy1

ĀÆy2ī.)

c) Let V=R2. Set

hx, yi=x1y1ā4x2y2,for x= (x1, x2), y = (y1, y2)āV.

d) Let V=P2(C) be the space of polynomials of degree at most two, having complex

coeļ¬cients. Let

hf1, f2i= 3f1(0)f2(0) + +2f1(i)f2(i), f1, f2āV.

e) Let Vbe an inner product space, with inner product hĀ·,Ā·i0. Suppose that T:VāVis a

linear transformation. Set

hx, yi=hx, yi0+hT(x), T (y)i0, x, y āV.

f) Let Vbe an inner product space, with inner product hĀ·,Ā·i0. Suppose that T:VāVis a

linear transformation that has the property hT(x), x i06=0whenever xāVand x6=0.

Set

hx, yi=hx, T (y)i0+hT(x), y i0, x, y āV.

2. Ā§6.1, #22.

3. Let hĀ·,Ā·i be the inner product on R3deļ¬ned by

h(x, y, z),(x0, y0, z0)i= 3xx0+ 2yy0+zz0.

Let Ī²={(1,1,0),(1,0,1),(0,1,1) }.

a) Use the Gram-Schmidt process to convert Ī²to an orthogonal basis (relative to the above

inner product).

b) Express the vector (0,2,ā1) as a linear combination of the vectors from the orthogonal

basis obtained in part a).

1

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