Problem 1

1. Solve the following system of linear equations using the method taught in class.

x₁ + 2x₂ x₃ = 8

2x₁ + 3x₂ + 9x₃ = 5

4x₁ 5x₂ 17x₃ = 7

3. Let v = (3,1,3)^T be a vector expressed in coordinates with respect to the standard basis of R³. Find the coordinates of this vector with respect to the basis

b .

Problem 2

Find conditions on such that following system of linear equations has (a) exactly one solution, (b) no solutions, or (c) an infinite number of solutions.

2x₁ 2x₂ + x₃ = 2

4x₁ 4x₂ + 12x₃ = 4

2x₁ + x₂ = 2

Problem 3

4. a) Determine whether the following vectors form a basis of R⁴. If not, obtain a basis by adding and/or removing vectors from the set.

v , v , v , v .

1. b) A matrix is called singular if the homogeneous linear system Av = 0 has a non-trivial solution v6= 0.

Prove that AB = 0 implies that at least one of the matrices is singular.