ML Prep

One reduction, four structural conclusions

Problem

For

A=[123246],b=[12],A=\begin{bmatrix}1&2&3\\2&4&6\end{bmatrix},\qquad b=\begin{bmatrix}1\\2\end{bmatrix},

row-reduce [A∣b][A\mid b]. Give the rank of AA, a basis for its column space, a basis for its nullspace, and the full solution of Ax=bAx=b. Verify rank-nullity.

Reveal answer or reference solution

Subtracting twice the first row from the second gives

[A∣b]∼[123∣1000∣0][A\mid b]\sim\begin{bmatrix}1&2&3&\mid&1\\0&0&0&\mid&0\end{bmatrix}

Thus rank⁡(A)=1\operatorname{rank}(A)=1. A pivot-column basis for the column space is {(1,2)T}\{(1,2)^{\mathsf T}\}. Writing x2=sx_2=s and x3=tx_3=t gives

x=(1−2s−3t,s,t)Tx=(1-2s-3t,s,t)^{\mathsf T}

For the homogeneous system, a nullspace basis is

{(−2,1,0)T,(−3,0,1)T}\{(-2,1,0)^{\mathsf T},(-3,0,1)^{\mathsf T}\}

The nullity is 22, so rank⁡(A)+nullity⁡(A)=1+2=3\operatorname{rank}(A)+\operatorname{nullity}(A)=1+2=3, the number of columns.

Outcome

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