Finding bases for a matrix's subspaces

Bases of Null Space

• We know Null(A) is the set of vectors that land on the origin after the transformation.

$$\text{Null}(A)=\{\vec{x}|A\vec{x}=0\}$$

• So we carry A to RREF to find values of $\vec{x}$ that satisfy $A\vec{x}=0$.
• After RREF, we can write $\vec{x}$ as a linear combination of two (or more) other vectors (num. of vectors depends on the num of free vars).
• Since every vector in Null(A) can be represented as a linear combination of those two vectors, those two vectors are the bases.
• These two vectors are linearly independent cuz we carried the matrix to (R)REF.

Bases of Column Space

• Column spaces is basically the set of linear combinations of the columns of A.
• The definition itself tells us what are we “linearly combining”–columns of A.
• We just need to ensure that the columns of A are linearly independent, then they can be bases.
• For L.I, check the number of pivots in the RREF of A, that’s the number of linearly independent vectors in the set.
• Since columns without pivots can be expressed as a linear combination of columns with pivots, we know we have LD.
• This relationship carries between the regular and RREF forms since row operations do not affect dependence between columns of a matrix
• Just pick the original columns that have pivots in the RREF version, that’s your basis set.

Bases of Row Space

• This is the set of linear combinations of the rows of A.
• Again, by definition, we know what vectors to look at.
• We just throw out the rows that are linearly dependent.
• Here, since row operations do not change the row space, our basis set can contain the RREF forms of the rows.