Linear Algebra, 2020a

222 Chapter Three. Maps Between Spaces
(b) {ae x + be 2x | a, b ∈ R}, B = 〈e x ,e 2x 〉
(c) {a + bx + ce x + dxe x | a, b, c, d ∈ R}, B = 〈1, x, e x ,xe x 〉
1.31 Find the range of the linear transformation of R 2 represented with respect to
the standard
( )
bases by
(
each matrix.
)
( )
1 0 0 0
a b
(a)
(b)
(c) a matrix of the form
0 0 3 2
2a 2b
̌ 1.32 Can one matrix represent two different linear maps? That is, can Rep B,D (h) =
RepˆB, ˆD (ĥ)?
1.33 Prove Theorem 1.5.
̌ 1.34 Example 1.10 shows how to represent rotation of all vectors in the plane through
an angle θ about the origin, with respect to the standard bases.
(a) Rotation of all vectors in three-space through an angle θ about the x-axis is a
transformation of R 3 . Represent it with respect to the standard bases. Arrange
the rotation so that to someone whose feet are at the origin and whose head is
at (1, 0, 0), the movement appears clockwise.
(b) Repeat the prior item, only rotate about the y-axis instead. (Put the person’s
head at ⃗e 2 .)
(c) Repeat, about the z-axis.
(d) Extend the prior item to R 4 .(Hint: we can restate ‘rotate about the z-axis’
as ‘rotate parallel to the xy-plane’.)
1.35 (Schur’s Triangularization Lemma)
(a) Let U be a subspace of V and fix bases B U ⊆ B V . What is the relationship
between the representation of a vector from U with respect to B U and the
representation of that vector (viewed as a member of V) with respect to B V ?
(b) What about maps?
(c) Fix a basis B = 〈⃗β 1 ,...,⃗β n 〉 for V and observe that the spans
[∅] ={⃗0} ⊂ [{⃗β 1 }] ⊂ [{⃗β 1 , ⃗β 2 }] ⊂ ··· ⊂ [B] =V
form a strictly increasing chain of subspaces. Show that for any linear map
h: V → W there is a chain W 0 = {⃗0} ⊆ W 1 ⊆···⊆W m = W of subspaces of W
such that
h([{⃗β 1 ,...,⃗β i }]) ⊆ W i
for each i.
(d) Conclude that for every linear map h: V → W there are bases B, D so the
matrix representing h with respect to B, D is upper-triangular (that is, each
entry h i,j with i>jis zero).
(e) Is an upper-triangular representation unique?