What you may be confusing yourself with is the column space vs. a basis for the column space. A basis is indeed a list of columns and for a reduced matrix such as the one you have a basis for the column space is given by taking exactly the pivot columns (as you have said).
What is basis of row space?
The nonzero rows of a matrix in reduced row echelon form are clearly independent and therefore will always form a basis for the row space of A. Thus the dimension of the row space of A is the number of leading 1’s in rref(A). Theorem: The row space of A is equal to the row space of rref(A).
How do you find the basis of a range space?
To find a basis for the range of T, remember that the columns of M span its range . Find the largest INDEPENDENT subset of the set of the columns of M . To find a basis for the range of T, remember that the columns of M span its range . Find the largest INDEPENDENT subset of the set of the columns of M .
What is a basis for the null space?
In general, if A is in RREF, then a basis for the nullspace of A can be built up by doing the following: For each free variable, set it to 1 and the rest of the free variables to zero and solve for the pivot variables. The resulting solution will give a vector to be included in the basis.
What is Col space?
The column space is all the possible vectors you can create by taking linear combinations of the given matrix. In the same way that a linear equation is not the same as a line, a column space is similar to the span, but not the same. The column space is the matrix version of a span.
What is the basis of a matrix?
When we look for the basis of the kernel of a matrix, we remove all the redundant column vectors from the kernel, and keep the linearly independent column vectors. Therefore, a basis is just a combination of all the linearly independent vectors.
Is row space equal to column space?
TRUE. The row space of A equals the column space of AT, which for this particular A equals the column space of -A. Since A and -A have the same fundamental subspaces by part (b) of the previous question, we conclude that the row space of A equals the column space of A.
