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# a4 - (b Is the union V ∪ W necessarily a subspace of< 3...

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ASSIGNMENT 4 EE441 KUMAR Fall 2007 Due: Oct. 1 (Monday, please note) 1. Identify geometrically, as clearly as you can, the subset of 3-dimensional (Euclidean) space < 3 that corresponds to the column space of the matrix A = 0 2 3 5 4 0 . 2. Let S be the subset of the (x,y)-plane (i.e., the subset of < 2 ) given by S = { ( x, y ) | x + 4 y = 13 } . Is S a subspace of the (x,y)-plane? 3. Identify geometrically, as clearly as you can, the subspaces of 3-dimensional (Euclidean) space < 3 that correspond (i) to the rowspace and (ii) to the nullspace of the matrix: A = 1 2 - 3 2 - 1 4 4 3 - 2 . Do you notice any relationship between the two ? 4. Consider two subspaces V and W of < 3 . (a) Is the intersection V W necessarily a subspace of < 3 ? (b) Is the union

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Unformatted text preview: (b) Is the union V ∪ W necessarily a subspace of < 3 ? Explain your answer. 5. Consider the linear system x + y-z = 2 x + 2 y + z = 3 x + y + ( k 2-5) z = k where k is an arbitrary constant. For which choice(s) of k does this system have a unique solution ? For which choice(s) of k does the system have inﬁnitely many solutions ? For which choice(s) of k is the system inconsistent ? 1 6. Does the subset W = { ( x,y ) | x,y, ∈ < , and x,y ≥ 0 or x,y ≤ } i.e., the union of the ﬁrst and third quadrants of the ( x,y )-plane < 2 , constitute a subspace (over the real numbers) of the ( x,y )-plane ? (i) True (ii) False 2...
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