 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2014/digital-systems-linear-algebra-introduction-to-formal-proof/5/d/solution

The pivot columns of the reduced row echelon form are columns 1 and 3. Take those columns of the original matrix, since elementary row operations need not preserve the original column space:

⎩ ⎨ ⎧ - 1 12, 325 ⎭ ⎬ ⎫ .

For an explicit spanning check, column 2 is

- 2 a_{1}

, column 4 is

- a_{1} + 2 a_{3}

, and column 5 is

3 a_{1} - 2 a_{3}

. The first two coordinates of the displayed columns form a

2 \times 2

- 5 \neq = 0

, proving their linear independence. They are therefore a basis of the column space.

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