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Lecture 1

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

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Fiona T Rahman

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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 ,·i defines an inner product on the vector space
V. If it is an inner product, prove that each of the four conditions in the definition of inner
product is satisfied. 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=Z1
f1(t)f2(t)dt+f1(0)f2(0), f1, f2V.
b) Let V=C2and A=52i
2i3. Set hx, yi=xAy,x,yC2. (Here, if y= (y1, y2),
then y=¯y1
c) Let V=R2. Set
hx, yi=x1y14x2y2,for x= (x1, x2), y = (y1, y2)V.
d) Let V=P2(C) be the space of polynomials of degree at most two, having complex
coefficients. Let
hf1, f2i= 3f1(0)f2(0) + +2f1(i)f2(i), f1, f2V.
e) Let Vbe an inner product space, with inner product ,·i0. Suppose that T:VVis a
linear transformation. Set
hx, yi=hx, yi0+hT(x), T (y)i0, x, y V.
f) Let Vbe an inner product space, with inner product ,·i0. Suppose that T:VVis a
linear transformation that has the property hT(x), x i06=0whenever xVand x6=0.
hx, yi=hx, T (y)i0+hT(x), y i0, x, y V.
2. §6.1, #22.
3. Let ,·i be the inner product on R3defined by
h(x, y, z),(x0, y0, z0)i= 3xx0+ 2yy0+zz0.
Let β={(1,1,0),(1,0,1),(0,1,1) }.
a) Use the Gram-Schmidt process to convert βto an orthogonal basis (relative to the above
inner product).
b) Express the vector (0,2,1) as a linear combination of the vectors from the orthogonal
basis obtained in part a).
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