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Linear Algebra 2270 Homework 11 a preparation for the last quiz on 08/05/2015 Problems: 1. Perform Gram-Schmidt orthogonalization of the following vectors: ⎡1⎤ ⎡−1⎤ ⎡−2⎤ ⎡2⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v1 = ⎢ 0 ⎥ , v2 = ⎢ 1 ⎥ , v3 = ⎢ 1 ⎥ , v4 = ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢−1⎥ ⎢3⎥ ⎢4⎥ ⎢4⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ And using it find matrices Q̃, R̃ such that matrix ⎡∣ ∣ ∣ ∣ ⎤⎥ ⎢ ⎥ ⎢ A = ⎢v1 v2 v3 v4 ⎥ = Q̃R̃ ⎥ ⎢ ⎢∣ ∣ ∣ ∣ ⎥⎦ ⎣ Use this factorization of A to find the solution set of the linear system: Ax = 0 2. Perform Gram-Schmidt orthonormalization of the following vectors: ⎡ √2 ⎤ ⎡ 1 √2⎤ ⎡−√2⎤ ⎡√2⎤ ⎢ ⎥ ⎢2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎥ , v2 = ⎢ √ 0 ⎥ , v3 = ⎢ √ 0 ⎥ , v4 = ⎢√1 ⎥ v1 = ⎢ √ ⎢ ⎥ ⎢3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢− 2⎥ ⎢ 2⎥ ⎢2 2⎥ ⎢ 2⎥ ⎣ ⎦ ⎣2 ⎦ ⎣ ⎦ ⎣ ⎦ And find QR decomposition of matrix ⎡∣ ∣ ∣ ∣ ⎤⎥ ⎢ ⎢ ⎥ A = ⎢v1 v2 v3 v4 ⎥ ⎢ ⎥ ⎢∣ ∣ ∣ ∣ ⎥⎦ ⎣ 1 3. (a) Consider space R2 with the dot product and a set W = {[ ]}. Find W ⊥ and (W ⊥ )⊥ −1 1 ]). Find W ⊥ and (W ⊥ )⊥ −1 ⎡ 1 ⎤ ⎡0⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (c) Consider space R3 with the dot product and a set W = {⎢−1⎥ , ⎢0⎥}. Find W ⊥ and (W ⊥ )⊥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢1⎥ ⎣ ⎦ ⎣ ⎦ (d) Let V be any inner product space. Let W be any subset of V , W ⊂ V . Prove that W ⊥ is a subspace. (b) Consider space R2 with the dot product and a set W = span([ (e) Let V be a finite dimensional inner product space. Let W be any subspace of V , W ≺ V . Prove that W = (W ⊥ )⊥ . Hint: Start by considering an orthonormal basis of W , enrich it to an orthonormal basis of V , show that the additional vectors form a basis of W ⊥ . 4. (a) Find a determinant of A by performing row or column operations (or both). [In additional notes posted on the website, we prove that one can perform row operations. The use of column operations have been justified in the lecture.] ⎡2 2 2 ⎤ ⎢ ⎥ ⎢ ⎥ A = ⎢1 0 1 ⎥ ⎢ ⎥ ⎢2 −2 1⎥ ⎣ ⎦ 1 (b) Find a determinant of A by performing expansions according to rows or columns. For 2 × 2 matrices you may use a formula given in the lecture. ⎡2 ⎢ ⎢1 ⎢ A=⎢ ⎢2 ⎢ ⎢3 ⎣ 2 2 0 1 0 1 0 −1 1⎤⎥ 2⎥⎥ ⎥ 0⎥⎥ 1⎥⎦ Hint: It might be a good idea to start with an expansion according to the second column. (c) Using all available methods (expansion according to a row or a column or row and column operations), find the determinant of the matrix: ⎡2 ⎢ ⎢2 ⎢ A=⎢ ⎢2 ⎢ ⎢3 ⎣ 3 2 1⎤⎥ 3 4 1⎥⎥ ⎥ 1 −1 0⎥⎥ 0 −1 1⎥⎦ Hint: It might be a good idea to start with operations: r2 + = (−1)r1 , r2 = 21 r2 and then an expansion according to one of the rows. 2