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A set of vectors is called a basis of (or a basis of a subspace of ) provided
that
(a)
(or )
(b)
is linearly independent.
Closed under addition
A set is said to be closed under addition if for each element and the sum is also in .
Closed under scalar multiplication
A set is said to be closed under scalar multiplication if for each element and for each
scalar the product is also in .
Column space of a matrix
Let be an matrix. The column space of , denoted by , is the subspace of spanned by
the columns of .
Coordinate vector with respect to a basis
Let be an ordered basis. Then the coordinate vector is the column vector such that .
Dimension
Let be a subspace of . The dimension of is the number, , of elements in any basis of .
We write
Null space of a matrix
Let be an matrix. The null space of , denoted by , is the set of all vectors in such
that . It is a subspace of .
Nullity of a matrix
Let be a matrix. The dimension of the null space of is called the nullity of
.
Ordered basis
A basis in which the elements appear in a specific fixed order. Establishing an order
is necessary because a coordinate vector with respect to a given basis relies on the
order in which the basis elements appear.
Rank of a matrix
Let be a matrix. The dimension of the row space of is called the rank of
.
Rank-Nullity theorem
Let be an matrix. Then
Row space of a matrix
Let be an matrix. The row space of , denoted by , is the subspace of spanned by
the rows of .
Subspace
Suppose that is a nonempty subset of that is closed under addition and closed under
scalar multiplication. Then is a subspace of .
2024-09-06 02:11:46
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)