# What is the basis of a row space?

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## What is a basis for a 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).

## What is the basis of column space?

A basis for the column space of a matrix A is the columns of A corresponding to columns of rref(A) that contain leading ones. The solution to Ax = 0 (which can be easily obtained from rref(A) by augmenting it with a column of zeros) will be an arbitrary linear combination of vectors.

## Is basis for row space unique?

If the matrix is further simplified to reduced row echelon form, then the resulting basis is uniquely determined by the row space.

## How do you find the basis for Col A?

Only the first two columns of “A” are pivot columns. Therefore, a basis for “Col A” is the set { , } of the first two columns of “A”. To find a basis for “Nul A”, solve . Thus, the vector: is a basis for “Nul A”.

## Can a basis be infinite?

Infinitely dimensional spaces

A space is infinitely dimensional, if it has no basis consisting of finitely many vectors. By Zorn Lemma (see here), every space has a basis, so an infinite dimensional space has a basis consisting of infinite number of vectors (sometimes even uncountable).

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## What is the dimension of the null space?

The dimension of the Null Space of a matrix is called the ”nullity” of the matrix. f(rx + sy) = rf(x) + sf(y), for all x,y ∈ V and r,s ∈ R. fA :Rm −→Rn which is given by: fA(x) = Ax, for x ∈ Rm .