Newberger Math 247 Spring 03
Homework solutions: Section 1.8 #26, 27, 31, Section 2.3 #34,36,37,38
Section 1.8 #26,27,31.
26. Let uand vbe linearly independent vectors in R3and let Pbe the plane
through u,vand 0. The parametric equation of Pis x=su+tv(with s
and tin R). Show that a linear transformation T:R3→R3maps Ponto a
plane through 0, a line through 0or onto just the origin 0in R3. What must
be true about T(u) and T(v) in order for the image of the plane Pto be a
plane?
We want to know the image of Punder T. Any point xon Pwill have the
form x=su+tvwhere sand tare real numbers. So T(x) = sT (u) + tT (v),
and T(P)is the collection of all vectors of the form sT (u)+ tT (v)where sand
tare any real number. Thus the image of Punder Tis Span{T(u), T (v)}.
Since there are three cases for the geometric description of the span of two
vectors in R3, there are three cases for the geometric description of the image
of Punder T. First {T(u), T (v)}could be independent, in which case, the
image of Punder Tis a plane. Second, {T(u), T (v)}could be dependent but
not both zero, in which case the image of Pis a line. Third, both vectors T(u)
and T(v)could be zero, in which case the image of Pis simply 0.
27. a. Show that the line through vectors pand qin Rnmay be written in
parametric form x= (1 −t)p+tq.
b. The line segment from pto pis the set of points of the form (1−t)p+tq
where 0 ≤t≥1. Show that a linear transformation Tmaps this line segment
onto a line segment or onto a single point.
a. The line through vectors pand qin Rnis parallel to the vector q−pand
through p. Thus it has parametric vector form (q−p)t+p= (1 −t)p+tq.
b. Apply the transformation to the line segment. We get
T((1 −t)p+tq) = (1 −t)T(p) + tT(q)
since Tis linear. Thus there are two cases. First if T(p) = T(q)then this
expression simplifies to T(p), which is a single point. Second if T(p)6=T(q),
then the expression describes a line segment parallel to the vector T(q)−T(p)
and through T(p).
31. Let T:Rn→Rnbe a linear transformation, and let {v1,v2,v3}be a
linearly dependent set in Rn. Explain why the set {T(v1), T (v2), T (v3)}is
linearly dependent.
Since {v1,v2,v3}is linearly dependent, the homogeneous equation
[v1,v2,v3]x=0
has non-trivial solutions. Let cbe one of these non-trivial solutions. Then
[v1,v2,v3]c=0,
so
c1v1+c2v2+c3v3=0.
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