Math 211 (A01) Typed Homework 2

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1. We know a system of linear equations can have 0, 1, or infinitely many solutions.
(a) Explain why a system of linear equations cannot have exactly 2 solutions.
(b) For a homogeneous system of linear equations, what are the possibilities for the number
of solutions? Explain, and make sure to define homogeneous system.
2. Suppose M is a homogeneous system of 4 linear equations and 3 variables. Let M be the
non-augmented matrix of coefficients.
(a) If rank(M) = 3, how many solutions does the system M have? (You do not need to
define reduced row echelon form, but include all other relevant definitions.)
(b) If rank(M) = 2, how many solutions does the system M have?
(c) Could rank(M) = 4? Explain.
3. Consider the equation E given by
x1 − x2 + x3 − x4 = 0
and let V =







x1
x2
x3
x4




∈ R
4
:




x1
x2
x3
x4




is a solution to E



. Find vectors ⃗v1, . . . , ⃗vn such that V =
span{⃗v1, . . . , ⃗vn}. Explain your process.
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