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I was going over some linear algebra notes and I was wondering after putting a matrix A into rref(A) form, do the L.I. column vectors of a matrix make up the span of the matrix (and the dimension of this would be the rank R(A) of the matrix right)? Also, would this be Im(A)? I'm just confused on how to get the span and rank of a matrix, and also very confused on what Im(A) is and how to get it. Thank you!

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Row rank of a matrix is the same as column rank, so for a square matrix, both the row vectors and column vectors will have the same span. The rank of the matrix is the number of L.I. row vectors and/or column vectors, which is the dimension of the space spanned by these vectors.

The image of A (Im(A)) is the set of outputs from applying the matrix (via matrix multiplication) to an input vector (thinking of the matrix as a linear map). A basis for the image can be obtained by applying the matrix to each element of a basis for the space that it acts on. The kernel (or null set) of A (ker(A)) is the set of input vectors that map to zero on application of the map represented by A.

Comparing the linear map A in multiple dimensions to a one-dimensional function, the kernel of A corresponds to the zeros of the function and the image of A corresponds to the range of the function.

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  • $\begingroup$ thanks for the reply! so for rank, do we need to do rref(A) first? Also, does it matter if we do column vectors or row vectors for the rank(A)? And just to clarify, rank(A) is just the amount of L.I column or row vectors there are in the matrix, and the actual span would be any linear combination of those vectors? Thank you so much again! $\endgroup$ – sjfklsdafjks Oct 10 '18 at 16:49
  • $\begingroup$ Your first sentence basically says that the row and column spaces of a matrix are the same (“have the same span“). That’s patently false. $\endgroup$ – amd Oct 10 '18 at 20:39

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