$30.00
1. Let T : R
n → R
m be a linear transformation.
(a) Show that the null space of T is a subspace of R
n.
(b) Show that the range of T is a subspace of R
m.
2. (a) For a 4 × 3 matrix M, must the column space of M be identical to the column space of
rref(M)?
(b) For a 3 × 3 matrix N with rank(N) = 3, must the column space of N be identical to the
column space of rref(N)? Can the assumption that rank(N) = 3 be dropped?
3. For a linear transformation L : R
3 → R
3
, we have the following information:
L
1
1
0
=
2
1
1
L
−1
1
0
=
−3
3
3
L
0
0
1
=
0
0
0
.
(a) Write down a matrix for L.
(b) Describe the range of L as a point, line, plane, or hyperplane and give a basis for the
range of L.
(c) Describe the null space of L as a point, line, plane, or hyperplane and give a basis for the
null space of L.
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