Linguist George Lakoff states that much of meaning in language is gr.pdf

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Linguist George Lakoff states that much of meaning in language is grounded in bodily metaphors (e.g., describing an attractive man or women as “hot”). Can you think of interesting examples of this? Solution Is it true that all of us, not just poets, speak in metaphors, whether we realize it or not? Is it perhaps even true that we live by metaphors? In Metaphors We Live By George Lakoff, a linguist, and Mark Johnson, a philosopher, suggest that metaphors not only make our thoughts more vivid and interesting but that they actually structure our perceptions and understanding. Thinking of marriage as a \"contract agreement,\" for example, leads to one set of expectations, while thinking of it as \"team play,\" \"a negotiated settlement,\" \"Russian roulette,\" \"an indissoluble merger,\" or \"a religious sacrament\" will carry different sets of expectations. When a government thinks of its enemies as \"turkeys or \"clowns\" it does not take them as serious threats, but if the are \"pawns\" in the hands of the communists, they are taken seriously indeed. Metaphors We Live By has led many readers to a new recognition of how profoundly metaphors not only shape our view of life in the present but set up the expectations that determine what life well be for us in the future. (from introduction in The Conscious Reader).

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Hi please complete the following with detailed working out Find the eigenvalues of C1 C1 C1 C1 C2 C3 C3 C3 C3 cn cn Hint: You may find that the identity A2 (c1 c2 cm) is useful. Solution COORDINATES, EIGENVALUES, AND EIGENVECTORS 1. Review of coordinates 1.1. The standard basis. The standard basis for R n is the basis E = {e1, e2, . . . en}, where ej is the vector with 1 in the j-th position and zeros elsewhere. 1.2. Coordinates with respect to a basis. Let B = {v1, v2, . . . vn} be a basis for R n . We know that for each vector x R n there are unique scalar “coordinates” c1, c2, . . . cn such that (1.1) x = c1v1 + c2v2 + . . . + cnvn We call the vector c R n having coordinates c1, c2 . . . cn the “coordinate vector” of x with respect to the basis B, and denote it by [x]B. 1.3. First examples. (a) [x]E = x for each x R n . (b) For any basis B = {v1, v2, . . . vn} of R n : [vj ]B = ej (1 j n). (c) [0]B = 0 for any basis B (the zero vector has the same representation with respect to any basis. 1.4. Matrix calculation of coordinates. Letting B = {v1, v2, . . . vn} be a basis for R n we can rewrite equation (1.1) in matrix form as (1.2) x = V [x]B where V is the matrix having the vector vj for its j-th column, i.e. V = [v1, v2, . . . vn]. . 1 2 EIGENVALUES/EIGENVECTORS V is invertible because its columns are linearly independent (see, for example, Summary 3.2.9(v) of our textbook, pp. 119–120), so we can compute the B coordinate representation of any vector x R n by solving (1.2) for [x]B: (1.3) [x]B = V 1x For example, if in R 2 we take B = {v1, v2} to be our basis, where v1 = 1 3 and v2 = 2 5 , then V = 1 2 3 5 and V 1 = 5 2 3 1 . Suppose, for instance, that x = 1 1 . Then [x]B = V 1x = 5 2 3 1 1 1 = 7 4 More generally, if x = x1 x2 is any vector in R n then the same calculation shows that [x]B = 5x1 + 2x2 3x1 x2 . 1.5. The matrix of a linear transformation with respect to a basis. Recall that each n × n matrix induces a linear transformation T : R n R n by means of matrix multiplication: (1.4) T(x) = Ax (x R n ). We refer to A as the “standard matrix” for T. The lesson of what’s to follow is that it’s not always the best matrix for T! For a basis B = {v1, v2, . . . vn} for R n we define (1.5) [T]B = [[T(v1)]B, [T(v2)]B . . . [T(vn)]B] , i.e. [T]B is the matrix that has the B-coordinate representation of T(vj ) as its j-th column (1 j n). We call [T]B the matrix of T with respect to the basis B. 1.6. Example: B = E, the standard basis. In this case [T]B = [Ae1, Ae2, . . . Aen], the “standard matrix” for T. EIGENVALUES/EIGENVECTORS 3 1.7. Geometric examples in R 2 . In this next group of examples, L is a line through the origin in the direction of a non-zero vector v R 2 , and w is a non-zero vector perpendicular to v. We take as our basis for R 2 , not the standard one, but instead B = {v, w}. (a) Suppose T : R 2 R 2 is the transformation of orthogonal projection onto L. Then T(v) = v and T(w) = 0, so [T]B = [[T(v)]B, [T(w)]B] = [[v]B, [0]B] = .

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Linguist George Lakoff states that much of meaning in language is gr.pdf

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Linguist George Lakoff states that much of meaning in language is gr.pdf