Let U and V be vector spaces over the same field F. We will create a...

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Let U and V be vector spaces over the same field F. We will create a new, larger, vector space, which contains both of U and V as subspaces. This new vector space is called the direct sum of U and V. Note that the textbook uses this terminology but in a more restricted setting. The direct sum of U and V is denoted by U OV. As a set, it is very simple: just the cartesian product U x V. The scalar multiplication is defined by 1. (u, v) = (Au, Av), X E F, u e U, v E V, and the vector addition is defined by (u, v) + (u', v') = (u + u', v + v'), u, u' e U, v, v' e V. Instead of writing (u, v), we sometimes use a different separator, and can write u O v for the vector in the direct sum. Or, identifying u with (u, 0) and v with (0, v), we might write simply u + v for (u, v). If we see a sum like this involving vectors from different vector spaces, there is no choice but to interpret it as a direct sum. Let X, Y be sets and Fa field. Prove that FALY F* OF. Let U and V be vector spaces over the same field F. We will create a new, larger, vector space, which contains both of U and V as subspaces. This new vector space is called the direct sum of U and V. Note that the textbook uses this terminology but in a more restricted setting. The direct sum of U and V is denoted by U OV. As a set, it is very simple: just the cartesian product U x V. The scalar multiplication is defined by 1. (u, v) = (Au, Av), X E F, u e U, v E V, and the vector addition is defined by (u, v) + (u', v') = (u + u', v + v'), u, u' e U, v, v' e V. Instead of writing (u, v), we sometimes use a different separator, and can write u O v for the vector in the direct sum. Or, identifying u with (u, 0) and v with (0, v), we might write simply u + v for (u, v). If we see a sum like this involving vectors from different vector spaces, there is no choice but to interpret it as a direct sum. Let X, Y be sets and Fa field. Prove that FALY F* OF.