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Is this right? The dimension of the space spanned by a matrix is the number of columns of the matrix. The dimension of a matrix is the number of rows of a matrix. The matrix exists in R^m, where m is the number of rows. Can someone give a clear definition of basis, span, and dimensions?

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basis is a set of vectors which 1. all of members are linearly independent and 2. all combinations of members can produce space.
any question about this?
the dimension of a space is the cardinal of basis set.

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what is the "cardinal"? were my first two statements accurate?
cardinal is the number of members of a set. your statements are not wrong... but hard to have an imagination in mind. at least for me. ( :d ) . by the definition i gave above. you can easily find out what you have to be looking for. or actually what a basis is.
and also....the matrix that u mentioned in ur statement should have some conditions... that fullfil this, you know them?
the columns should be independent?
exactly...!!! which is condition 1 in basis definition. and second condition in definition produce the space of your matrix. but generally if the space be whatever... any vectors ( even objects) the importance of condition 2 becomes more clear. i hope i didn't confused you. need more explanation?
welcome :) good luck :)

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