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hw1 - STAT424 Spring 2010 Homework#1 Homework 1 Due Tuesday...

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STAT424 Spring 2010 Homework #1 Jan 26, 2010 Homework 1 Due: Tuesday, Feb 2, 2010 1) Questions involve the following matrices: A = 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 1 ; B = 1 2 0 1 2 0 0 1 0 0 1 1 ; D = 1 2 1 2 2 5 0 0 ; E = 1 0 1 0 3 0 0 2 . (a) Which of the matrices have linearly independent columns? (b) Which of the following equalities are valid: C ( A ) = C ( B ), C ( A ) = C ( A, D ), C ( A, B ) = C ( D, E )? (c) Give the ranks of the matrices. (d) Give an orthonormal basis for C ( A ). (e) Find C ( A ) and C ( D ) (with respect to R 4 ). 2) Write the following models in matrix notation and find the dimension of C ( X ). (a) One-way ANOVA model y ij = μ + α i + ij , where i = 1 : 3, and j = 1 : 2 when i = 1 , 2 and j = 1 : 3 when i = 3. (b) Same as (a) but with a constraint that α 1 = α 2 . (c) Same as (a) but with a consraint that α 1 + α 2 + α 3 = 0. (d) Two-way ANOVA without interaction y ijk = μ + α i + β j + ijk , where i = 1 , 2 , 3 , j = 1 , 2, and k = 1.
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