Common Mistakes Using Math Terminology
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The word ``solution''
Equations and inequalities can have solutions (see the handout ``What is a
solution?''). However,
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A matrix does not have a solution.
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A vector does not have a solution.
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A set of vectors does not have a solution.
The word ``equation''
An equation is a sentence with an equals sign in it.
Expressions that lack ``='' signs are not equations.
In particular:
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A vector is not an equation.
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A sum of vectors is not an equation.
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A linear combination of vectors is not an equation.
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A matrix is not the same thing as a system of equations. (One can
associate a matrix to a system of equations, but they are not
the same animal.)
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Linear (in)dependence
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Linear dependence and linear independence are terms that
apply only to a set of vectors in a vector space. In particular:
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A 3x3 matrix cannot be a linearly independent set in
R3 because it does not live in
R3. (See "Sets of vectors" below)
- A subspace cannot be linearly independent.
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Linear combinations
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A set of vectors is not the same thing as a linear
combination of vectors.
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A set of linear combinations of vectors is not the same thing
as a linear combination of vectors. (A linear combination of vectors
is a single vector, not an infinite set of vectors. A set of linear
combinations of vectors is usually an infinite set of vectors. This
is why a
span is not a linear combination.)
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One-to-one and Onto
Only functions can be one-to-one or onto. In particular:
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An equation cannot be one-to-one or onto.
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A matrix cannot be one-to-one or onto.
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A vector cannot be one-to-one or onto.
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Span
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"Span" and words constructed from it occur as different parts
of speech, each with a different meaning. The sentence
"A spanning set spans its span"
is both grammatically correct and true.
- "A spans B" is not the same statement as "B spans A". For
example, a subspace does not span a set of three vectors, but
a set of three vectors spans a subspace.
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A span is not the same thing as a spanning set.
- A span is not a linear combination. (A span is a
set of linear combinations and usually contains infinitely many
vectors. A linear combination is a single vector.)
- A 3x3 matrix does not span R3. Only a set
of vectors in R3 can span R3.
(A matrix is not the same thing as a set of vectors.)
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Subspaces
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A set of three vectors is not a subspace.
(Here "three" could be any finite number greater than or equal to
one, except that the single-vector set {0} is a
subspace.)
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A subspace is not a basis.
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A basis is not a
subspace .
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A subspace is not a matrix.
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A matrix is not a
subspace .
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A subspace cannot be linearly independent.
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A subspace cannot be consistent or inconsistent. Those
words apply only to equations and systems of equations.
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Basis
- A basis { v1, v2, ... ,
vp} is not the same as the matrix [
v1 v2 ... vp].
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A basis is not a vector space.
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A basis is not a subspace .
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A subspace is not a basis.
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A basis is not the same thing as the vector space or subspace
it's a basis of. In particular:
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A basis of a column space is not the same thing as that column
space.
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A basis of a null space is not the same thing as that null
space.
- A basis is not the same thing as a linear combination
of vectors.
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A basis is not the same thing as a sum of vectors.
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A basis is not the same thing as a solution set to a system
of linear equations.
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A basis is not isomorphic to a vector space. Only two vector spaces
can be isomorphic.
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Sets of vectors
- A matrix is not the same thing as a set of vectors. The
set { v1, v2, ... , vp}
is not the same as the matrix [ v1 v2
... vp].
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Elements
An element of a set is an object in that set. In particular:
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The components (or entries) of a vector in
Rn are real numbers. These components are not
elements of Rn (unless n=1); they are
elements of R.
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Miscellaneous
- A vector is not the same thing as a vector space. (A
vector does not equal a set of vectors.)
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A plus sign does not mean the same thing as a comma, or the same thing
as the word ``and''. Do not put plus signs between elements of a list
of vectors unless you really mean to add the vectors.
This page was last modified by D. Groisser on Apr. 11, 1999.