Mathematics for Computer Scientists 2 (COMP1045 UNMC) (SPM1 24-25)
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Another implication of Steinitz' Lemma is that if V has a basis with n vectors, then any linearly independent set of n vectors in V is also a:
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A linear mapping T: V -> W is injective (one-to-one) if and only if:
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Which axiom of a vector space guarantees the existence of a vector 0 in E such that for every vector u in E, u + 0 = u?
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The existence of a neutral elements (zero vectors) in a vector space V is defined as:
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Grassmann's Formula relates the dimensions of two subspaces W1 and W2 of a finite-dimensional vector space V as:
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The linear mapping T: R^2 -> R^3 defined by T(x, y) = (x + y, 2x - y, y) has a matrix representation (with respect to the standard bases) given by:
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A key implication of Steinitz' Lemma is that if a vector space V is spanned by n vectors, then any linearly independent set in V can have at most:
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A linear mapping T: V -> W is surjective (onto) if and only if:
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If dim(W1) = 3, dim(W2) = 4, and dim(W1 ∩ W2) = 2, then dim(W1 + W2) is:
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The dimension of a vector space is the maximum number of:
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