Unformatted text preview: plane related to each other? Problem 3. Let A be an n × m matrix. Assume that there exists an m × n matrix B such that B A = I n . Show that rank( A ) = m . Must A necessarily be invertible? Problem 4. Consider the linear transformation T : R 3 → R 3 deﬁned by T ( ~x ) = A~x in terms of the 3 × 3 matrix A = 1 7 2 3 6 36 2 6 23 . a. Show that T is an orthogonal transformation. b. Find the matrix of the inverse of T . Problem 5. Let V be a linear space with inner product h·i . Prove the Law of Cosines : Show that if θ is the angle between f,g ∈ V , then  fg  2 =  f  2 +  g  22 cos θ  f  g  ....
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 Spring '08
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 Linear Algebra, Algebra, Vector Space, web site, Linear map

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