Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 9E / a / ii

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- Solution ii

Solution

Row reduction of the matrices whose columns are the displayed spanning vectors gives

dim U = 3, dim W = 3.

Row-reducing the matrix formed from all six columns gives

dim (U + W) = 4.

The dimension formula for a sum of subspaces now yields

dim (U \cap W) = dim U + dim W - dim (U + W) = 3 + 3 - 4 = 2 .

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