T : V \to Z

dim V = dim ker T + dim im T .

U \oplus W

consists of pairs

(u, w)

with componentwise vector-space operations. The canonical inclusions are

i (u) = (u, 0), j (w) = (0, w) .

For subspaces

U, W \leq V

q : U \oplus W ⟶ U + W, q (u, w) = u + w .

This is surjective, and

ker q = {(x, - x) : x \in U \cap W} = im l .

(U \oplus W) / im l ≅ U + W \leq V .

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.

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

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