Problem 1
a) 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
b) 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 α such that following system of linear equations has (a) exactly one solution, (b) no solutions, or (c) an infinite number of solutions.
2x1 − 2x2 + αx3 = −2
4x1 − 4x2 + 12x3 = −4
2x1 + αx2 = 2
Problem 3
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 .
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.
