 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/digital-systems-linear-algebra-introduction-to-formal-proof/5/d/solution

The pivot columns in the row echelon form are columns 1 and 2. Take the corresponding columns of the original matrix:

Col (A) has basis ⎩ ⎨ ⎧ 1220, 1301 ⎭ ⎬ ⎫ .

For a direct check, call these columns

c_{1}, c_{2}

. The other columns satisfy

c_{3} = c_{1} - c_{2}

and

c_{4} = c_{1} + c_{2}

, so they are in the same linear span. Also

c_{1}, c_{2}

are linearly independent: their fourth entries force the coefficient of

c_{2}

to vanish, then their first entries force the other coefficient to vanish. Elementary row operations preserve dependence relations among columns, but generally change the column space itself, which is why the original columns must be used.

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