Linear Algebra Properties

Let A be an m × n matrix and x and y two n-vectors. If Ax = 5 and
Ay = 3, what is A(x − y)?

• 2

• 15

• 8

• -2

Let V be a vector space. Which of the following are not true?
(a) The multiplication of any two vectors in V is a vector in V .
(b) There is a vector ~x ∈ V such that for all vectors ~y ∈ V , ~x + ~y = ~y.
(c) The sum of any two vectors in V is a vector in V .
(d) For any real number c ∈ R and any vector ~x ∈ V , c~v is a vector
in V .

• (a) The multiplication of any two vectors in V is a vector in V .

• 
  (b) There is a vector ~x ∈ V such that for all vectors ~y ∈ V , ~x + ~y = ~y.

• 
  (c) The sum of any two vectors in V is a vector in V
  

• d) For any real number c ∈ R and any vector ~x ∈ V , c~v is a vector
  in V .

Let A, B be invertible 50 × 50 matrices. Which of the following is not
true?

• (a) The inverse of A is unique.

• (b) (AB)^−1 = A−1^B^−1.

• (c) (BT )^−1 = (B^−1)^T .

• (d) The inverse of B can be written as a product of elementary matrices.

Consider two subspaces A and B of the larger vector space V . If V =
A + B, dim A = m, dim B = n, and dim(A ∩ B) = k, then what is the
dimension of V ?

• (a) m + n

• b) m − n − k

• (c) m + n − k

• (d) mn − k

Consider two subspaces A and B of the larger vector space V . If V =
A ⊕ B, dim A = m, and dim V = n, then what is the dimension of B?

• (a) mn

• (b) m − n

• (c) m + n

• (d) n − m

(True of False) Let U be an orthogonal matrix. That is, U is square
and U>U = UU> = I. If ~x ⊥ ~y, then U~x ⊥ U~y. (The symbol ⊥ means
“perpindicular.”)

• True

• False

(True of False) Let U be an orthogonal matrix. That is, U is square
and U>U = UU> = I. If U~x = ~y, then the length of ~x equals the
length of ~y.

• True

• False

Suppose P is a projection matrix and Pc its complementary projection
matrix. If P~x = ~y, then what is Pc~y?

• x

• 0

• y

Suppose both A and B are n × n matrices. If det A = 5 and det B = 2,
then what is det (AB)?

• 7

• 10

• -3

• 3