Past exam of the mathematical methods I course of the University of Oxford / 2021 / 8 / c

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- Solution c

Solution

The dimension formula for a sum of subspaces gives

dim (U + V) + dim (U \cap V) = 4.

Since the sum lies in

R^{3}

and each plane has dimension two, the possibilities are

(dim (U + V), dim (U \cap V)) = (2, 2) or (3, 1) .

The first case means the planes coincide; two distinct planes through the origin intersect in a line.

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