# Adv Alegbra HW Solutions 130 - n × n matrix A is Gaussian...

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130 (iii) Gaussian equivalent matrices have the same column space. Solution. False. (iv) The matrix A = h 10 0 01 1 00 1 i is nonsingular. Solution. True. (v) Every nonsingular matrix over a f eld is a product of elementary matrices. Solution. True. (vi) If A is an m × n matrix, then Ro w( A T ) = Col ( A ) . Solution. True. 4.24 (i) Prove that a list v 1 ,...,v m in a vector space V is linearly inde- pendent if and only if it spans an m -dimensional subspace of V . Solution. (ii) Determine whether the list v 1 = ( 1 , 1 , 1 , 2 ), v 2 = ( 2 , 2 , 3 , 1 ), v 3 = ( 1 , 1 , 0 , 5 ) in k 4 is linearly independent. Solution. Let A be the matrix whose rows are the given vectors, and see whether rank ( A ) = m . 4.25 Do the vectors v 1 = ( 1 , 4 , 3 ) , v 2 = ( 1 , 2 , 0 ) , v 3 = ( 2 , 2 , 3 ) span k 3 ? Solution. Yes, because rank ( A ) = 3. 4.26 (i) Prove that every n × n row reduced echelon matrix is triangular. Solution. Absent. (ii) Use Theorem 4.39 to prove that every
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Unformatted text preview: n × n matrix A is Gaussian equivalent to a triangular matrix. Solution. Absent. 4.27 Prove that Ax = β is consistent if and only if β lies in the column space of A . Solution. If γ ∈ k m , then A γ is a linear combination of the columns of A . 4.28 If A is an n × n nonsingular matrix, prove that any system Ax = b has a unique solution, namely, x = A − 1 b . Solution. Absent. 4.29 Let α 1 , . . . , α n be the columns of an m × n matrix A over a f eld k , and let β ∈ k m . (i) Prove that β ∈ h α 1 , . . . , α n i if and only if the inhomogeneous system Ax = β has a solution. Solution. Absent....
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