Dear Researchers,
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
https://www.researchgate.net/post/What_is_the_relation_between_rank_span_and_cardinality_in_linear_algebra
I guess you can interprete the rows as vectors, the you can decide what space is covered by these vectors.
Then you do the same for the other matrix.
ex.
01
10
will span the whole plane.
1 0
20
will only span the x axis.
thank you for your explanations Juan Weisz and Peter Breuer
To sum up, what I learnt from your answers is that if two matrices span the same space, they are not necessarily the equal, am I right, please?
In terms of vectors of the matrix space, it means that A=cB for some c in the field ground. This equality has many consequence, for example, their row space, minimal polynomial are equal, etc....
Dear Bakhtiar Karim
Your question is not well-posed.
What do you mean by a space spanned by a matrix?
If you fix the question, you may get good answers.
Regards
If you have two matrices of the same dimensions, say, A,B \in R^(m \times n), they are vectors in the linear space R^(m \times n). Span of matrix A is {tA: t \in R}. Two matrices A & B span the same spaces iff A=\alpha B for some nonzero \alpha. To make the situation meaningful, we assume A and B are nonzero matrices. This is my understanding of the problem.
Ok, Let us consider this way. Take a m(rows) by n(columns) matrix. If you now consider the matrix as coordinates of m numbers of points in n-dimensional space, then it would be natural to assume that, if these points are plotted in an n-dimensional space, then it is less likely that any (n-1) dimensional hyperplane, or in general, any lower-dimensional space existing "inside" that n-dimensional space would not fit all the "points". But this is not necessarily always true. Take, for example, any plane passing through our well-known and common 3D space, plot any number of points on that plane (which would obviously lie on that 3d space), and note down their 3D coordinates into a matrix. Although there is need for 3-dimensional data to note down these coordinates, all the points can well be fitted through a 2-dimensional subspace that is part of 3-dimensional space.
"span" of a matrix, once consistent custom of identifying data points and coordinates are established, indicates the minimum "dimension" of space needed for all the data points. For example [1 0; 2 0] has span of 1 dimension, because although 2D space can be used to formally represent them, 1D space (e.g. the x-axis in case of xy plane) suffices to represent the data points. Same goes for [1 1; 3 3] matrix that has span of 1 dimension, but the line on xy plane would have equation of x-y=0. Of course, numerically the "direction" of two matrices do not match up, but rotating the data points (implied in the matrix) in space would perfectly align them.
Thus 'span' is a measure of (hyper)dimensionality of (data points implied in) the matrix.
thank you so much for your contribution and a special thank to Peter Breuer for his clarifications
Article Small-span Hermitian matrices over quadratic integer rings
Preprint Matrices and Spans and internalization and enrichment in tricategories
Preprint Distances to the Span of Sparse Random Matrices, with Applic...
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