Column Space – The vector space that is spanned by all the C(A) columns in the matrix (A), can also be by row notation written as R(Transpose(A))
If a vector is contained in the column space of a certain matrix, then that vector is a column vector and the coefficients/weights needed to calculate that vector are post multiplying the matrix and are in the same dimensions as the contained vector.
Row Space -The vector space that is spanned by all the R(A) rows in the matrix (A),can also be written by column notation as C(Transpose(A))
If a vector is contained in the column space of a certain matrix, then that vector is a row vector and the coefficients/weights needed to calculate that vector are pre multiplying the matrix and are in the same dimensions as the contained vector.
One of the main questions in Statistics is:
Is Some vector contained in the column space/row space of a certain matrix and if not, than how close can we get to the column space/row space ?
This is solved by the minimizing distance/magnitude as close to zero as possible using the Linear Least Squared Algorithm.
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