Lecture 9, basis - Harvard Math Department
... equivalent to the problem to find a basis for the kernel of the matrix which has the vectors w ~ i in its rows. FINDING A BASIS FOR THE IMAGE. Bring the m × n matrix A into the form rref(A). Call a column a pivot column, if it contains a leading 1. The corresponding set of column vectors of the orig ...
... equivalent to the problem to find a basis for the kernel of the matrix which has the vectors w ~ i in its rows. FINDING A BASIS FOR THE IMAGE. Bring the m × n matrix A into the form rref(A). Call a column a pivot column, if it contains a leading 1. The corresponding set of column vectors of the orig ...
Notes on Vector Spaces
... 1. If V is any vector space, V is a subspace of V ; the subset consisting of the zero vector alone is a subspace of V , called the zero subspace of V . 2. If F n , the set of n-tuples (x1 , x2 , ..., xn ) with x1 = 0 is a subspace; however, the set of n-tuples with x1 = 1 + x2 is not a subspace (n ≥ ...
... 1. If V is any vector space, V is a subspace of V ; the subset consisting of the zero vector alone is a subspace of V , called the zero subspace of V . 2. If F n , the set of n-tuples (x1 , x2 , ..., xn ) with x1 = 0 is a subspace; however, the set of n-tuples with x1 = 1 + x2 is not a subspace (n ≥ ...
Quaternionic groups November 5, 2014
... do with symplectic groups. This is true, and will be explained in (0.4b). There is some disagreement about whether symplectic groups should be labelled with 2n (the dimension of the vector space) or n (the “rank,” which we’ll talk about eventually). I think that for the noncompact groups 2n wins, bu ...
... do with symplectic groups. This is true, and will be explained in (0.4b). There is some disagreement about whether symplectic groups should be labelled with 2n (the dimension of the vector space) or n (the “rank,” which we’ll talk about eventually). I think that for the noncompact groups 2n wins, bu ...
THE GERTRUDE STEIN THEOREM As we saw in the TQFT course
... As we saw in the TQFT course, Frobenius algebras are important structures in the study of TQFTs. However, as it may be desirable to connect with other areas of mathematics, a broader set of definitions would be useful in showing that non-categorical structures are Frobenius algebras. So, we have thr ...
... As we saw in the TQFT course, Frobenius algebras are important structures in the study of TQFTs. However, as it may be desirable to connect with other areas of mathematics, a broader set of definitions would be useful in showing that non-categorical structures are Frobenius algebras. So, we have thr ...
Lie Algebra Cohomology
... Then A is a left g-module and x ◦ a is K-linear in x and a. Note also that by the universal property of U g the map ρ induces a unique algebra homomorphism ρ1 : U g → EndK A, thus making A in a left U gmodule. Conversely, if A is a left U g-module, so that we have a structure map σ : U g → EndK A, i ...
... Then A is a left g-module and x ◦ a is K-linear in x and a. Note also that by the universal property of U g the map ρ induces a unique algebra homomorphism ρ1 : U g → EndK A, thus making A in a left U gmodule. Conversely, if A is a left U g-module, so that we have a structure map σ : U g → EndK A, i ...
Note
... The null space of A N(A) is the same as the null space of U. If the system A.x = 0 is reduced to U.x = 0 then none of the solutions (which constitute the null space) is changed. It has dimension n – r. If homogeneous solutions to U.x = 0 are found, they constitute a basis for N(A). The column space ...
... The null space of A N(A) is the same as the null space of U. If the system A.x = 0 is reduced to U.x = 0 then none of the solutions (which constitute the null space) is changed. It has dimension n – r. If homogeneous solutions to U.x = 0 are found, they constitute a basis for N(A). The column space ...
Semidefinite and Second Order Cone Programming Seminar Fall 2001 Lecture 10
... of minimum polynomial coincides with the linear algebraic notion we are already familiar with. Also, a matrix X is regular if its characteristic polynomial (defined as F(t) = Det(tI − X)) coincides with its minimum polynomial. (From linear algebra we know that the minimum polynomial of a matrix divi ...
... of minimum polynomial coincides with the linear algebraic notion we are already familiar with. Also, a matrix X is regular if its characteristic polynomial (defined as F(t) = Det(tI − X)) coincides with its minimum polynomial. (From linear algebra we know that the minimum polynomial of a matrix divi ...
Lecture 15: Dimension
... B = {~v1 , . . . , ~vn } ⊂ X is a basis if two conditions are satisfied: B is linear independent meaning that c1~v1 + ... + cn~vn = 0 implies c1 = . . . = cn = 0. Then B span X: ~v ∈ X then ~v = a1~v1 + . . . + an~vn . The spanning condition for a basis assures that there are enough vectors to repre ...
... B = {~v1 , . . . , ~vn } ⊂ X is a basis if two conditions are satisfied: B is linear independent meaning that c1~v1 + ... + cn~vn = 0 implies c1 = . . . = cn = 0. Then B span X: ~v ∈ X then ~v = a1~v1 + . . . + an~vn . The spanning condition for a basis assures that there are enough vectors to repre ...
DEFORMATION THEORY
... M. DOUBEK, M. MARKL AND P. ZIMA Abstract. First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformatio ...
... M. DOUBEK, M. MARKL AND P. ZIMA Abstract. First three sections of this overview paper cover classical topics of deformation theory of associative algebras and necessary background material. We then analyze algebraic structures of the Hochschild cohomology and describe the relation between deformatio ...
I
... Segre variety and its secant varieties The set of all rank-1 hypermatrices is known as the Segre variety in algebraic geometry. It is a closed set (in both the Euclidean and Zariski sense) as it can be described algebraically: Seg(Rl , Rm , Rn ) = {A ∈ Rl×m×n | A = u ⊗ v ⊗ w} = {A ∈ Rl×m×n | ai1 i2 ...
... Segre variety and its secant varieties The set of all rank-1 hypermatrices is known as the Segre variety in algebraic geometry. It is a closed set (in both the Euclidean and Zariski sense) as it can be described algebraically: Seg(Rl , Rm , Rn ) = {A ∈ Rl×m×n | A = u ⊗ v ⊗ w} = {A ∈ Rl×m×n | ai1 i2 ...
Segments and Angles
... Vector fields can be used in some areas of physics: • As fluid flows through a pipe, the velocity vector V can be found at any location within the pipe. • The gravity created by a mass creates a vector field where every vector points at the mass, and the magnitudes get smaller as points are chosen ...
... Vector fields can be used in some areas of physics: • As fluid flows through a pipe, the velocity vector V can be found at any location within the pipe. • The gravity created by a mass creates a vector field where every vector points at the mass, and the magnitudes get smaller as points are chosen ...
Lecture 16 - Math TAMU
... Definition. An n×n matrix B is said to be similar to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they ...
... Definition. An n×n matrix B is said to be similar to an n×n matrix A if B = S −1 AS for some nonsingular n×n matrix S. Remark. Two n×n matrices are similar if and only if they represent the same linear operator on Rn with respect to different bases. Theorem If A and B are similar matrices then they ...
Phil 312: Intermediate Logic, Precept 7.
... of Boolean algebra. Recall: a Boolean algebra B is a set together with a unary operation ¬, two binary operations ∧ and ∨, and designated elements 0 ∈ B and 1 ∈ B which satisfy the equations which Prof. Halvorson wrote on the board yesterday. (Note that perhaps it would be better to refer to these o ...
... of Boolean algebra. Recall: a Boolean algebra B is a set together with a unary operation ¬, two binary operations ∧ and ∨, and designated elements 0 ∈ B and 1 ∈ B which satisfy the equations which Prof. Halvorson wrote on the board yesterday. (Note that perhaps it would be better to refer to these o ...
Similarity - U.I.U.C. Math
... vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X: v = c1 x1 + · · · + cn xn , where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the coordinate list of v wrt X (with respect to X). Of course, every vector w ∈ W has a coordinate list ...
... vector v ∈ V can be written in exactly one way as a linear combination of the basis vectors of X: v = c1 x1 + · · · + cn xn , where all ci ∈ R; we call the n × 1 column vector (c1 , . . . , cn )> the coordinate list of v wrt X (with respect to X). Of course, every vector w ∈ W has a coordinate list ...
Orthogonal Projections and Least Squares
... subspace of W . (2) If S is a subspace of the inner product space V , then S ⊥ is also a subspace of V . Proof: (1.) Note that 0+0 = 0 is in U ⊕V . Now suppose w1 , w2 ∈ U ⊕V , then w1 = u1 +v1 and w2 = u2 +v2 with ui ∈ U and vi ∈ V and w1 + w2 = (u1 + v1 ) + (u2 + v2 ) = (u1 + u2 ) + (v1 + v2 ). Si ...
... subspace of W . (2) If S is a subspace of the inner product space V , then S ⊥ is also a subspace of V . Proof: (1.) Note that 0+0 = 0 is in U ⊕V . Now suppose w1 , w2 ∈ U ⊕V , then w1 = u1 +v1 and w2 = u2 +v2 with ui ∈ U and vi ∈ V and w1 + w2 = (u1 + v1 ) + (u2 + v2 ) = (u1 + u2 ) + (v1 + v2 ). Si ...
General vector spaces ® So far we have seen special spaces of
... as Col(A) (or C(A)), is the set of all linear combinations of the columns of A. If A = [a1 · · · an], then Col(A) = span{a1, ..., an} The column space of an m×n matrix A is a subspace of Rm. ä A vector in Col(A) can be written as Ax for some x [Recall that Ax stands for a linear combination of the c ...
... as Col(A) (or C(A)), is the set of all linear combinations of the columns of A. If A = [a1 · · · an], then Col(A) = span{a1, ..., an} The column space of an m×n matrix A is a subspace of Rm. ä A vector in Col(A) can be written as Ax for some x [Recall that Ax stands for a linear combination of the c ...
Math 416 Midterm 1. Solutions. Question 1, Version 1. Let us define
... Question 3, Version 2. Show that S = {1, x − 2, x + 1, x2 − 1, x2 } spans P2 (R). Write down a subset of S that is a basis for P2 (R). Solution: A really quick way to show that S spans P2 (R) is to note that 1 and x2 are both in S, and clearly x ∈ Span(S), since (x + 1) + (x − 2) + 1 = x. Since {1, ...
... Question 3, Version 2. Show that S = {1, x − 2, x + 1, x2 − 1, x2 } spans P2 (R). Write down a subset of S that is a basis for P2 (R). Solution: A really quick way to show that S spans P2 (R) is to note that 1 and x2 are both in S, and clearly x ∈ Span(S), since (x + 1) + (x − 2) + 1 = x. Since {1, ...
math21b.review1.spring01
... leading 1’s are leading variables, but when dealing with an augmented matrix a leading 1 in the last column does not represent any variable, but indicates an inconsistent system; if all the variables are leading 1’s then there is a unique solution to the equation, but if any are nonleading variables ...
... leading 1’s are leading variables, but when dealing with an augmented matrix a leading 1 in the last column does not represent any variable, but indicates an inconsistent system; if all the variables are leading 1’s then there is a unique solution to the equation, but if any are nonleading variables ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.