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
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.
Apologize for the misuse of terminologies. I think the column vectors v1 and v2 index the matrix A. Please let me know if I am correct.
No, I would not use the word "index" in this case. Why can't you simply write that v1 is the first row of A, and that v2 is the second row of A? Or perhaps, "the first two rows of A are v1 and v2 (written as row vectors)."
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.
Dear Jeffrey L Stuart Maged G. Bin-Saad
In linear algebra, what does actually mean when it is said that two matrices span the same space?
For example, if matrix A spans the same space as matrix B, is that mean A=B? or
what does this information (i.e., spanning the same space) tell about the relationship between A and B?
I appreciate your response and your clarifications
Hello Dr. Jeffrey L Stuart ,
I have one question which is related to our previous discussion under this section.
Are eigenvectors the orthonormal basis vectors? Please let me know.
Hello Bakhtiar Karim ,
Firstly a vector space is seen as a set of vectors that is closed under an operation(usually the addition operation). A Vector space is categorized as a group which in general means a set of elements which are closed under a binary operation.
If there is a set of vectors S, and let us say there are vector x1 and x2 which are a part of the set S then x3 =a1x1 + a2x2 is the third element vector that is also part of S. a1 and a2 are scalar coefficients. A vector space usually has properties of additive, multiplicative, inverse, and commutative.
A span is a set of linear combinations of the vectors that are a part of the vector space S.
If matrix an m-by-n A is partitioned into column vectors a1...am and they belong to the vector space S then x1a1+x2a2+...xmam will be the span(a1....am) x1 and x2 so on will be their scalar coefficients.
Similarly with the matrix B. Matrix A may not be the same as B as the span may even vary with the scalar coefficients since they could either belong to real or complex domain.
Please let me know your comments and also please do correct me if I am wrong.
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