A/ ( vectors are bold; scalars are italics.)

1) A set V upon which the operations of addition and scalar multiplication
have been defined is called a vector space if and only if these 10 axioms are true.

1) if , then . (closure under addition)

2) u + v = v + u (addition is commutative)

3) u + (v + w) = (u + v) + w (addition is associative)

4) there is a 0 in V such that 0 + u = u + 0 = u (additive identity)

5) for all , there exists (– u) such that u + (– u) = 0. (additive inverse)

6) for any real number k, . (closure under real number scalar multiplication)

7) k(u + v) = ku + kv. (scalar on a vector sum distributive property)

8) (k + l)u = ku + lu. (scalar sum on a vector distributive property)

9) k(lu) = (kl)u. (commutative property under scalar multiplication)

10) 1u = u. (multiplicative identity = 1, a scalar)

Test these sets to see if they are vector spaces. If they're not, list all the axioms that fail.

a) V is the subset of R³ in which u₂ = 0. Addition is standard,
scalar multiplication is defined as ku =

b) V is the subset of R³ in which u₃ = 0 with standard addition and scalar multiplication.

c) V is the subset of R² with standard scalar multiplication vector addition is:

d) V is the subset of 2 × 2 matrices with a₁₁ = 0, a₁₂, a₂₁, and a₂₂ are elements of R.

(12)

2) Determine conditions on x₁, x₂ and x₃ so that the vector

(4)

3)

Given that

determine if x belongs to the space spanned by u, v, and w.

(4)

B/ 1) Test these sets for linear independence. Explain your answer.

a)	b)
c)	d)

(8)

2) Find an equation for the plane spanned by:

(4)

C/

1) a) Find a basis for the solution space of

b) What dimension is this space?

c) Prove that it is a subspace of R⁵.

(8)

2)

a) Find a basis for the row space of A.

b) Find a basis for the column space of A.

c) What is the rank of A?

d) Prove that if M is a 3 x 5 matrix, the column vectors of M are linearly dependent.

(10)

TOTAL (50)

.

( Linear Algebra MathRoom Index )

(all content © MathRoom Learning Service; 2004 - ).