 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/4/b/i/solution

Let

c_{1} = (1, 1, 1)^{T}

and

c_{2} = (2, - 2, 6)^{T}

. The remaining columns are

3 c_{1}

and

2 c_{2}

, so these two vectors span the column space. They are linearly independent: the determinant of their first two coordinates is

- 4 \neq = 0

. Therefore ${(1, 1, 1)^{T}, (2, - 2, 6)^{T}}$ is a basis of the range, and the matrix rank is two.

Equivalently the range is the plane

y_{3} = 2 y_{1} - y_{2}

: both displayed basis vectors satisfy this equation, and their two-dimensional span fills that plane.

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