Linear Algebra, 2020a

Section III. Computing Linear Maps 229
(b) The second member?
(c) Where is a general vector from the domain (a vector with components x and
y) mapped? That is, what transformation of R 2 is represented with respect to
B, D by this matrix?
2.17 Consider a homomorphism h: R 2 → R 2 represented with respect to the standard
bases E 2 , E 2 by this matrix.
Find the
(
image under
(
h of each
(
vector.
)
2 0 −1
(a) (b) (c)
3)
1)
1
( ) 1 3
2 4
2.18 What transformation of F = {a cos θ + b sin θ | a, b ∈ R} is represented with
respect to B = 〈cos θ − sin θ, sin θ〉 and D = 〈cos θ + sin θ, cos θ〉 by this matrix?
( ) 0 0
1 0
̌ 2.19 Decide whether 1 + 2x is in the range of the map from R 3 to P 2 represented
with respect to E 3 and 〈1, 1 + x 2 ,x〉 by this matrix.
⎛ ⎞
1 3 0
⎝0 1 0⎠
1 0 1
2.20 Find the map that this matrix represents with respect to B, B.
( ) ( ( 2 1
1 1
B = 〈 , 〉
−1 0
0)
1)
2.21 Example 2.11 gives a matrix that is singular and is therefore associated with
maps that are singular. We cannot state the action of the associated map g on
domain elements ⃗v ∈ V, because do not know the domain V or codomain W or
the starting and ending bases B and D. But we can compute what happens to the
representations Rep B,D (⃗v).
(a) Find the set of column vectors representing the members of the null space of
any map g represented by this matrix.
(b) Find the nullity of any such map g.
(c) Find the set of column vectors representing the members of the range space
of any map g represented by the matrix.
(d) Find the rank of any such map g.
(e) Check that rank plus nullity equals the dimension of the domain.
̌ 2.22 Take each matrix to represent h: R m → R n with respect to the standard bases.
For each (i) state m and n. Then set up an augmented matrix with the given
matrix on the left and a vector representing a range space element on the right
(e.g., if the codomain is R 3 then in the right-hand column put the three entries a,
b, and c). Perform Gauss-Jordan reduction. Use that to (ii) find R(h) and rank(h)
(and state whether the underlying map is onto), and (iii) find N (h) and nullity(h)
(and state whether the underlying map is one-to-one).
( ) 2 1
(a)
−1 3