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What is the relation between rank span and cardinality in linear algebra

What is the relation between rank span and cardinality in linear algebra

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What is the relation between rank, span and cardinality

in linear algebra?

Considering a matrix A which has vectors v1=[1;0;0] and v2=[1;1;0] i.e. this matrix A is

spanned by the vectors v1 and v2.

The rank of this matrix A is 2. Going by the definition of the rank of a matrix it means

the number of independent vectors or the dimension of the row space.

Seeing A={v1,v2} with a cardinality of 2 an we say that the cardinality is the same as

the rank of the matrix which in turn means that it gives the number of independent

vectors spanning A

Solution:

In linear algebra, we reserve the word "span" for the construction of vector subspaces. A matrix

is not "spanned" by a set of vectors. We might describe a matrix in terms of its row vectors (or

column vectors) as being "built" with a particular finite set of vectors. In your example, it

happens that the set of rows is independent, so the rank is equal to the number of rows in the

matrix. In general, this does not happen.

Let A be an m x n matrix with rank r. Then 0 =< r =< min{m,n}. r = 0 exactly when A is the m x n

all zero matrix. Equivalently, the row space of A is the trivial subspace of F^n where F is usually

either the set of all real numbers or all complex numbers.

Suppose r is positive. A has (or perhaps, is built with) m row vectors from F^n. The SET of

nonzero rows in the reduced row echelon matrix RREF(A) is a basis for the row space of A,

which is an r-dimensional subspace of F^n. That is, the row space of A is spanned by the set of

nonzero rows of RREF(A). The dimension of the row space of A equals the cardinality of the set

of nonzero rows in RREF(A). Since the rank of A is defined to be the cardinality of the set of

nonzero rows in RREF(A), we see that the rank of the matrix A equals the dimension of the row

space of A.

another solution:

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or

spanned) by its columns. This corresponds to the maximal number of linearly independent

columns of A. This, in turn, is identical to the dimension of the vector space spanned by its

rows.

By definition, two sets are of the same cardinality if there exists a one-to-one correspondence

between their elements. For a finite set, the cardinality is the number of its elements. ... For

example, Z and R are infinite sets of different cardinalities while Z and Q are infinite sets of the

same cardinality.

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What is the relation between rank, span and cardinality in linear algebra? Considering a matrix A which has vectors v1=[1;0;0] and v2=[1;1;0] i.e. this matrix A is spanned by the vectors v1 and v2. The rank of this matrix A is 2. Going by the definition of the rank of a matrix it means the number of independent vectors or the dimension of the row space. Seeing A={v1,v2} with a cardinality of 2 an we say that the cardinality is the same as the rank of the matrix which in turn means that it gives the number of independent vectors spanning A Solution: In linear algebra, we reserve the word "span ...
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