Linear Algebra, 2020a

Section II. Homomorphisms 197
1.21 Is (perpendicular) projection from R 3 to the xz-plane a homomorphism? Projection
to the yz-plane? To the x-axis? The y-axis? The z-axis? Projection to the
origin?
1.22 Verify that each map is a homomorphism.
(a) h: P 3 → R 2 given by
(b) f: R 2 → R 3 given by
ax 2 + bx + c ↦→
( ) a + b
a + c
⎛ ⎞
( 0
x
↦→ ⎝x − y⎠
y)
3y
1.23 Show that, while the maps from Example 1.3 preserve linear operations, they
are not isomorphisms.
1.24 Is an identity map a linear transformation?
̌ 1.25 Stating that a function is ‘linear’ is different than stating that its graph is a
line.
(a) The function f 1 : R → R given by f 1 (x) =2x − 1 has a graph that is a line.
Show that it is not a linear function.
(b) The function f 2 : R 2 → R given by
( x
↦→ x + 2y
y)
does not have a graph that is a line. Show that it is a linear function.
̌ 1.26 Part of the definition of a linear function is that it respects addition. Does a
linear function respect subtraction?
1.27 Assume that h is a linear transformation of V and that 〈⃗β 1 ,...,⃗β n 〉 is a basis
of V. Prove each statement.
(a) If h(⃗β i )=⃗0 for each basis vector then h is the zero map.
(b) If h(⃗β i )=⃗β i for each basis vector then h is the identity map.
(c) If there is a scalar r such that h(⃗β i )=r·⃗β i for each basis vector then h(⃗v) =r·⃗v
for all vectors in V.
1.28 Consider the vector space R + where vector addition and scalar multiplication
are not the ones inherited from R but rather are these: a + b is the product of
a and b, and r · a is the r-th power of a. (This was shown to be a vector space
in an earlier exercise.) Verify that the natural logarithm map ln: R + → R is a
homomorphism between these two spaces. Is it an isomorphism?
1.29 Consider this transformation of the plane R 2 .
( ( )
x x/2
↦→
y)
y/3
Find the image under this map of this ellipse.
( x
{ | (x
y)
2 /4)+(y 2 /9) =1}
̌ 1.30 Imagine a rope wound around the earth’s equator so that it fits snugly (suppose
that the earth is a sphere). How much extra rope must we add so that around the
entire world the rope will now be six feet off the ground?