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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.

1

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