Prove the so-called Dimension of a Sum Theorem. Assume that U and U are two subspaces...

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• Prove the so-called "Dimension of a Sum" Theorem. Assume that U₁ and U₂ are two subspaces of a finite-dimensional vector space. Prove in a clear manner that dime (U₁+U₂) = dimR (U₁) + dime (U₂) - dime (U₁U₂). In particular, as you discuss the proof, consider the following points in your writing: a) Specify the set of working assumptions of the Theorem (i.e., the hypotheses). b) Specify the set of statements to be proven in the Theorem (i.e., the theses). c) Emphasize exactly where in the proof the working assumptions are being used. d) Emphasize the role played by the concepts of basis and dimension in the proof. • Prove the so-called "Dimension of a Sum" Theorem. Assume that U₁ and U₂ are two subspaces of a finite-dimensional vector space. Prove in a clear manner that dime (U₁+U₂) = dimR (U₁) + dime (U₂) - dime (U₁U₂). In particular, as you discuss the proof, consider the following points in your writing: a) Specify the set of working assumptions of the Theorem (i.e., the hypotheses). b) Specify the set of statements to be proven in the Theorem (i.e., the theses). c) Emphasize exactly where in the proof the working assumptions are being used. d) Emphasize the role played by the concepts of basis and dimension in the proof.

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To prove the Dimension of a Sum Theorem we start by noting that any vector in the sum U1 U2 can be written as a linear combination of vectors in U1 and U2 Therefore we can consider the sets of basis v... View the full answer