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Lecture 1

# MAT247H1 Lecture Notes - Lecture 1: Kolmogorov Space, Linear Map, Linear Combination

Department
Mathematics
Course Code
MAT247H1
Professor
Fiona T Rahman
Lecture
1

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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=īZ1
0
f1(t)f2(t)dtī+f1(0)f2(0), f1, f2āV.
b) Let V=C2and A=ī5ā2i
āā2i3ī. Set hx, yi=xAyā,x,yāC2. (Here, if y= (y1, y2),
then yā=īĀÆy1
ĀÆy2ī.)
c) Let V=R2. Set
hx, yi=x1y1ā4x2y2,for x= (x1, x2), y = (y1, y2)āV.
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, f2āV.
e) Let Vbe an inner product space, with inner product hĀ·,Ā·i0. Suppose that T:VāVis a
linear transformation. Set
hx, yi=hx, yi0+hT(x), T (y)i0, x, y āV.
f) Let Vbe an inner product space, with inner product hĀ·,Ā·i0. Suppose that T:VāVis a
linear transformation that has the property hT(x), x i06=0whenever xāVand x6=0.
Set
hx, yi=hx, T (y)i0+hT(x), y i0, x, y āV.
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,ā1) as a linear combination of the vectors from the orthogonal
basis obtained in part a).
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