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MathLinear Algebra

Rank Nullity Theorem: Find Null-Space and Column-Space Bases

Find bases for the null and column spaces of two MIT exercise matrices, compute rank and nullity, and verify that their sum is the number of columns.

Question

For each matrix below, give a basis of its null space NS(A) and a basis of its column space CS(A). Verify the rank-nullity theorem in each case.

(a)

A=(1111)A=\begin{pmatrix}1&1\\1&1\end{pmatrix}

(b)

A=(12342345)A=\begin{pmatrix}1&2&3&4\\2&3&4&5\end{pmatrix}

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Evidence boundary

This page answers both matrices in the exercises on page 11 of the MIT 18.03 Section 7.2 notes. The matrices were read from the original handwritten PDF. Bases and row reductions are derived here over the real numbers; equivalent bases may differ from the ones displayed.