LINEAR ALGEBRA EXAM 2 REVIEW
(CHAPTERS 4, 5) QUESTIONS WITH
CORRECT ANSWERS
Vector Space - Answer-A vector space is a nonempty set V of objects, called vectors,
on which are defined two operations, called addition and multiplication by scalars(real
numbers), subject to the ten axioms (or rules) listed below.1 The axioms must hold for
all vectors u, v, and w in V and for all scalars c and d.
1. The sum of u and v, denoted by u + v, is in V .
2. u + v = v + u.
3. (u + v) + w = u + (v + w)
4. There is a zero vector 0 in V such that u + 0 = u.
5. For each u in V , there is a vector -u in V such that u + (-u) = 0.
6. The scalar multiple of u by c, denoted by cu, is in V .
7. c(u + v) = cu + cv.
8. (c + d)u = cu + du.
9. c(du) = (cd)u
10. 1u = u
Note that: - Answer-For each u in V and scalar c,
0u = 0 (1)
c0 = 0 (2)
-u =(-1)u (3)
Subspace of a vector space - Answer-A subspace of a vector space V is a subset H of
V that has three properties:
a. The zero vector of V is in H. 2
b. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in
H.
c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c,
the vector cu is in H.
Theorem 1 - Answer-If v1; : : : ; vp are in a vector space V , then Span fv1; : : : ; vpg is a
subspace of V .
Null Space of a Matrix - Answer-The null space of an m n matrix A, written as Nul A, is
the set of all solutions of the homogeneous equation Ax = 0. In set notation, Nul A = {x :
x is in R n and Ax = 0}
Theorem 2 - Answer-The null space of an m X n matrix A is a subspace of Rn .
Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear
equations in n unknowns is a subspace of R^n.
(CHAPTERS 4, 5) QUESTIONS WITH
CORRECT ANSWERS
Vector Space - Answer-A vector space is a nonempty set V of objects, called vectors,
on which are defined two operations, called addition and multiplication by scalars(real
numbers), subject to the ten axioms (or rules) listed below.1 The axioms must hold for
all vectors u, v, and w in V and for all scalars c and d.
1. The sum of u and v, denoted by u + v, is in V .
2. u + v = v + u.
3. (u + v) + w = u + (v + w)
4. There is a zero vector 0 in V such that u + 0 = u.
5. For each u in V , there is a vector -u in V such that u + (-u) = 0.
6. The scalar multiple of u by c, denoted by cu, is in V .
7. c(u + v) = cu + cv.
8. (c + d)u = cu + du.
9. c(du) = (cd)u
10. 1u = u
Note that: - Answer-For each u in V and scalar c,
0u = 0 (1)
c0 = 0 (2)
-u =(-1)u (3)
Subspace of a vector space - Answer-A subspace of a vector space V is a subset H of
V that has three properties:
a. The zero vector of V is in H. 2
b. H is closed under vector addition. That is, for each u and v in H, the sum u + v is in
H.
c. H is closed under multiplication by scalars. That is, for each u in H and each scalar c,
the vector cu is in H.
Theorem 1 - Answer-If v1; : : : ; vp are in a vector space V , then Span fv1; : : : ; vpg is a
subspace of V .
Null Space of a Matrix - Answer-The null space of an m n matrix A, written as Nul A, is
the set of all solutions of the homogeneous equation Ax = 0. In set notation, Nul A = {x :
x is in R n and Ax = 0}
Theorem 2 - Answer-The null space of an m X n matrix A is a subspace of Rn .
Equivalently, the set of all solutions to a system Ax = 0 of m homogeneous linear
equations in n unknowns is a subspace of R^n.
