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Advanced Calculus: Problems and applications to science and by Hugo. Rossi

By Hugo. Rossi

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Sample text

Pair that the standard basis vectors But dim R3 = 3, so this is 3 also. = Independence Repeating the definition once again, dimension is the minimum number of linear space. " In such phrases as "there is a four parameter family of curves," "two vectors it takes to span a are involved," allusion is being made to a dimension-like notion. Now, if we try to pin down this notion mathematic ally and specify the concept of independence in the linear space context, it turns out to be precisely the requirement for a spanning set of vectors to be minimal.

Rv 6 V. e reR,\eV implies a set of vectors in R", the linear span of S, denoted [5] is the set of all vectors of the form c'-Vi -\ 1- ckvk with v, VjeS. (iii) The dimension of a linear subspace V of R" is the minimum number of vectors whose linear span is V. Linear Span Having we now had better just given the intuitively loaded word "dimension" a definition, hope that it suits our preconception of that notion. It does that in R3 : line is one dimensional since it is the linear span of but one is two dimensional because we need that many vectors plane to span it.

Is the equation (I + A)(I + B)=I +A+B possible (with nonzero AandB)? 22. An n X n Show that diagonal matrices, AB = matrix to have (a/) is said to be diagonal if a/ 0 for i =j. matrices commute; that is, if A and B are diagonal Give necessary and sufficient conditions for a diagonal matrix A BA. an inverse. = = Linear Functions 1. 4 In the last section Ax we saw that the equation b = be solved can of R" Subspaces for b's restricted just by certain linear equations and that the equation might have some degrees of freedom.

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