Ch-3 Vector Spaces and Subspaces-1-web
... real numbers whether they are written as row vectors or column vectors. If we wish to think of row vectors as n × 1 matrices, column vectors as 1 × n matrices, and consider matrix multiplication, then we must distinguish between row and column vectors. If we wish the scalars to be the complex number ...
... real numbers whether they are written as row vectors or column vectors. If we wish to think of row vectors as n × 1 matrices, column vectors as 1 × n matrices, and consider matrix multiplication, then we must distinguish between row and column vectors. If we wish the scalars to be the complex number ...
Vector Spaces in Quantum Mechanics
... which can be easily shown to be orthonormal. For instance û1 = ...
... which can be easily shown to be orthonormal. For instance û1 = ...
Notes
... (a) F = (x2 − y)i + 2xj, C1 , C2 both run from (−1, 0) to (1, 0) with C1 along the x-axis and C2 along the parabola y = 1 − x2 . (b) F = xyi − x2 j, C: quarter circle running from (0, 1) to (1, 0). (c) F = yi − xj, C: the triangle with vertices (0, 0), (0, 1), (1, 0) oriented clockwise. (d) F = yi, ...
... (a) F = (x2 − y)i + 2xj, C1 , C2 both run from (−1, 0) to (1, 0) with C1 along the x-axis and C2 along the parabola y = 1 − x2 . (b) F = xyi − x2 j, C: quarter circle running from (0, 1) to (1, 0). (c) F = yi − xj, C: the triangle with vertices (0, 0), (0, 1), (1, 0) oriented clockwise. (d) F = yi, ...
Vector Algebra and Vector Fields Part 1. Vector Algebra. Part 2
... This also follows directly from the commutative law for the components. The associative law has no meaning in relation to the scalar product. For instance, if we take the scalar product ~a ~b, then this is a scalar, and it is meaningless to form its dot product with a third vector. The scalar prod ...
... This also follows directly from the commutative law for the components. The associative law has no meaning in relation to the scalar product. For instance, if we take the scalar product ~a ~b, then this is a scalar, and it is meaningless to form its dot product with a third vector. The scalar prod ...
Notes on the Dual Space Let V be a vector space over a field F. The
... There is a canonical mapping R of a vector space V into its second dual V ∗∗ = (V ∗ )∗ defined by R(v) = v ∗∗ where v ∗∗ (φ) = φ(v). The proof of the linearity of v ∗∗ and R are left to the reader. If R(v) = 0 we have φ(v) = 0 for all φ ∈ V ∗ . If v 6= 0 then it can be completed to a basis B of V . ...
... There is a canonical mapping R of a vector space V into its second dual V ∗∗ = (V ∗ )∗ defined by R(v) = v ∗∗ where v ∗∗ (φ) = φ(v). The proof of the linearity of v ∗∗ and R are left to the reader. If R(v) = 0 we have φ(v) = 0 for all φ ∈ V ∗ . If v 6= 0 then it can be completed to a basis B of V . ...
Math 412: Problem Set 2 (due 21/9/2016) Practice P1 Let {V i∈I be
... such that ϕ ◦ σi0 = σi (hint: construct ϕ by assumption, and a reverse map using the existence part of 5(b); to see that the composition is the identity use the uniqueness of the assumption and of 5(b), depending on the order of composition). D. Now let P be a vector space equipped with maps πi0 : P ...
... such that ϕ ◦ σi0 = σi (hint: construct ϕ by assumption, and a reverse map using the existence part of 5(b); to see that the composition is the identity use the uniqueness of the assumption and of 5(b), depending on the order of composition). D. Now let P be a vector space equipped with maps πi0 : P ...
Second midterm solutions
... We see that α ∈ V and α ∈ W , so α ∈ V ∩ W . Since u1 , . . . , uj is a basis for V ∩ W , α can be written in a unique way as a linear combination of u1 , . . . , uj . This same expression must be the unique way of writing α as a linear combination of the basis u1 , . . . , uj , w1 , . . . , w for ...
... We see that α ∈ V and α ∈ W , so α ∈ V ∩ W . Since u1 , . . . , uj is a basis for V ∩ W , α can be written in a unique way as a linear combination of u1 , . . . , uj . This same expression must be the unique way of writing α as a linear combination of the basis u1 , . . . , uj , w1 , . . . , w for ...
