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# a5 - the system has a unique solution What does the...

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ASSIGNMENT 5 EE441 KUMAR Fall 2007 Due: Monday, Oct. 8 1. How many possible patterns can you find (like the (5 × 8) example matrix U in Fig. 2.3 , page 79 of the text) for (2 × 3) row-reduced echelon matrices. Entries to the right of the pivots are irrelevant. 2. Find conditions on b 1 , b 2 , b 3 such that the system of linear equations 2 x + 2 y + z + 7 w = b 1 x + y + 2 z + 8 w = b 2 x + y + 2 w = b 3 will have at least one solution. Given that b 1 = - 6 , b 2 = 0, choose b 3 such that the system of linear equations has at least one solution (it turns out that this is possible in this case). Then for this choice of b 1 , b 2 , b 3 , find all solutions to the system of linear equations. Follow the steps outlined in class: (i) first identify the augmented matrix; then reduce the left-hand side of this augmented matrix to echelon form; find the condition(s) for existence of a solution. Then choose b 3 appropriately, identify a particular solution and the nullspace of the coefficient matrix and finally, write down an expression for the set of all solutions. 3. Consider a linear system of four equations with three unknowns.

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Unformatted text preview: the system has a unique solution. What does the reduced-row echelon form of the coeﬃcient matrix of this system look like ? Use ﬂ to denote a pivot and * to denote an entry that may or may not be zero. 4. Identify precisely all conditions on a,b,c,d under which the columns of A = a a a a a b b b a b c c a b c d are linearly independent. Before you begin, explain your approach in a brief sentence. 5. If the vectors u , v and w of a vector space V are linearly independent, does it necessarily follow that the vectors u + v , v + w and w + u will always be linearly independent ? 1 6. Does the vector x = -1-1-3 belong to the subspace of 3-dimensional space < 3 spanned by the vectors a = 1 1 2 and b = 2 1 4 ? 2...
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a5 - the system has a unique solution What does the...

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