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Existence
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Uniqueness
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Equations
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Are there
solutions to
Ax=b
for a given b?
(Is the aug. system
[A b] consistent?)
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Are there
solutions to
Ax=b
for all b?
(Is the aug. system
[A b] consistent
for any b?)
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Is the
trivial solution x=0
to Ax=0
the only one?
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Echelon form
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Is the
augmentation column
pivotless?
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Do all rows
have pivots
in the coefficient part?
(If yes, then
no pivot can fall in the
augmentation column,
no matter what b is.)
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Do all
coefficient columns
have pivots?
(A pivotless
coefficient column
means a free variable.)
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Vectors
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Is b in the
span(cols of A)?
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Do cols of A span
the whole space?
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Are cols of A
linearly independent?
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Linear map T
(T(x)=Ax)
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Is b in the
range(T)?
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Is T onto?
(Does range(T)
= codomain(T)?)
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Is T 1-1?
(Does ker(T) = 0?)
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