Document
... “line perpendicular to a plane” • A line is perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. ...
... “line perpendicular to a plane” • A line is perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. ...
1. To find the image length of a 4-foot tall object in a spherical mirror
... The length of a rectangle is 2 units greater than the width, w. What is the area of the rectangle? A ...
... The length of a rectangle is 2 units greater than the width, w. What is the area of the rectangle? A ...
Purely Algebraic Results in Spectral Theory
... has a solution x ∈ A, given that a, b, c ∈ A. Here we assume that A is a non-commutative algebra and that F is an arbitrary field of scalars, not necessarily algebraically closed. As usual one could consider A to be the algebra of all n × n matrices with entries in F. The key condition for the exist ...
... has a solution x ∈ A, given that a, b, c ∈ A. Here we assume that A is a non-commutative algebra and that F is an arbitrary field of scalars, not necessarily algebraically closed. As usual one could consider A to be the algebra of all n × n matrices with entries in F. The key condition for the exist ...
Solutions
... 1. Let k be an algebraically closed field, and let Z ⊆ A3k be the set Z = {(t, t2 , t3 ) | t ∈ k}. Find generators for the ideal I(Z) in k[x, y, z], and show that the ring A(Z) = k[x, y, z]/I(Z) is isomorphic to a polynomial ring in one variable over k. (Z is called the twisted cubic curve in A3 .) ...
... 1. Let k be an algebraically closed field, and let Z ⊆ A3k be the set Z = {(t, t2 , t3 ) | t ∈ k}. Find generators for the ideal I(Z) in k[x, y, z], and show that the ring A(Z) = k[x, y, z]/I(Z) is isomorphic to a polynomial ring in one variable over k. (Z is called the twisted cubic curve in A3 .) ...
Polynomials over finite fields
... 1 dimensional affine subspaces = lines Lx,y = { x+λy | λ Є F } 2 dimensional affine subspaces = planes Px,y,z = { x+λy+μz | λ,μ Є F } n-1 dimensional affine subspaces = hyperplanes ...
... 1 dimensional affine subspaces = lines Lx,y = { x+λy | λ Є F } 2 dimensional affine subspaces = planes Px,y,z = { x+λy+μz | λ,μ Є F } n-1 dimensional affine subspaces = hyperplanes ...
Algebraic Properties
... The three properties are used to simplify algebraic expressions. The _________________________________________________ is used to expand groups and remove parentheses from expressions. The _________________________________________________ is used to change the order of the numbers so that like terms ...
... The three properties are used to simplify algebraic expressions. The _________________________________________________ is used to expand groups and remove parentheses from expressions. The _________________________________________________ is used to change the order of the numbers so that like terms ...
Block 2
... supplementary angles. Multiply two vulgar fractions where the denominator of one and the numerator of the other are equal. ...
... supplementary angles. Multiply two vulgar fractions where the denominator of one and the numerator of the other are equal. ...
10 Rings
... a ring (field). For instance 2 and 3 are algebraic integers of degree 2, but 2 + 3 is not. It will be algebraic of degree 4 however. In fact, one has the following. Proposition 10.7. The algebraic integers form a subring of C. The algebraic numbers form a subfield of C. The proof is somewhat technic ...
... a ring (field). For instance 2 and 3 are algebraic integers of degree 2, but 2 + 3 is not. It will be algebraic of degree 4 however. In fact, one has the following. Proposition 10.7. The algebraic integers form a subring of C. The algebraic numbers form a subfield of C. The proof is somewhat technic ...
aa4.pdf
... Find the eigenvalues of the map u and describe the corresponding eigenspaces. 4. Let G → GL(V ) be a finite dimensional continuous representation of a compact group G and let S ⊂ V be a convex G-stable subset. Show that there exists a G-fixed point s ∈ S. 5. Let k be a field. For each a ∈ k× and eac ...
... Find the eigenvalues of the map u and describe the corresponding eigenspaces. 4. Let G → GL(V ) be a finite dimensional continuous representation of a compact group G and let S ⊂ V be a convex G-stable subset. Show that there exists a G-fixed point s ∈ S. 5. Let k be a field. For each a ∈ k× and eac ...
Graduate Qualifying Exam in Algebra School of Mathematics, University of Minnesota
... (b) (9 points) Now let K be a field and let R be the factor ring R = K[X]/(X 2(X +1)). Determine how many ideals of R are projective as R-modules, giving generators for each such ideal. 8. (12 points) Let α be a complex number which is algebraic over Q and let L be a subfield of C which is a splitti ...
... (b) (9 points) Now let K be a field and let R be the factor ring R = K[X]/(X 2(X +1)). Determine how many ideals of R are projective as R-modules, giving generators for each such ideal. 8. (12 points) Let α be a complex number which is algebraic over Q and let L be a subfield of C which is a splitti ...
Study Guide
... Understand that an algebraic expression is simply a way to show a number if we don’t know all its parts Be able to translate words into algebraic expressions Understand that each term in an algebraic expression (2xy) has a “what” (in this case xy) and a “how many” (in this case 2) Understand the dif ...
... Understand that an algebraic expression is simply a way to show a number if we don’t know all its parts Be able to translate words into algebraic expressions Understand that each term in an algebraic expression (2xy) has a “what” (in this case xy) and a “how many” (in this case 2) Understand the dif ...
PM 464
... I(An (|)) = 0 if | is infinite. 3. S ⊆ I(V (S)) for any set of polynomials S ⊆ |[x 1 , . . . , x n ]. X ⊆ V (I(X )) for any set of points X ⊆ An (|). 4. V (I(V (S))) = V (S) for any set of polynomials S ⊆ |[x 1 , . . . , x n ]. I(V (I(X ))) = I(X ) for any set of points X ⊆ An (|). Remark. Equality ...
... I(An (|)) = 0 if | is infinite. 3. S ⊆ I(V (S)) for any set of polynomials S ⊆ |[x 1 , . . . , x n ]. X ⊆ V (I(X )) for any set of points X ⊆ An (|). 4. V (I(V (S))) = V (S) for any set of polynomials S ⊆ |[x 1 , . . . , x n ]. I(V (I(X ))) = I(X ) for any set of points X ⊆ An (|). Remark. Equality ...
Details about the ACCUPLACER EXAM
... ACCUPLACER tests use a multiple-choice format. There’s no time limit on the tests, so you can focus on doing your best to demonstrate your skills. ACCUPLACER uses the latest computer-adaptive technology. Questions are presented based on your individual skill level. Your response to each question dri ...
... ACCUPLACER tests use a multiple-choice format. There’s no time limit on the tests, so you can focus on doing your best to demonstrate your skills. ACCUPLACER uses the latest computer-adaptive technology. Questions are presented based on your individual skill level. Your response to each question dri ...
s principle
... their categories differ from others . ( Such considerations will be important in order to carry out Grothendieck ’ s 1 973 program [ 2 ] for s implifying the foundations of algebraic geometry . ) An axiomatic theory often captures more examples than originally intended . In the present case not only ...
... their categories differ from others . ( Such considerations will be important in order to carry out Grothendieck ’ s 1 973 program [ 2 ] for s implifying the foundations of algebraic geometry . ) An axiomatic theory often captures more examples than originally intended . In the present case not only ...
Math 210B. Homework 4 1. (i) If X is a topological space and a
... and 1 in R = k[X, Y ]/(X(X − 1)(X − λ)) where λ ∈ k − {0, 1}, and determine the associated decomposition of R as a direct product in each case. Draw pictures. (iii) If Z ⊂ k n is an affine algebraic set, prove every point has a connected neighborhood (so all connected components are open) and interp ...
... and 1 in R = k[X, Y ]/(X(X − 1)(X − λ)) where λ ∈ k − {0, 1}, and determine the associated decomposition of R as a direct product in each case. Draw pictures. (iii) If Z ⊂ k n is an affine algebraic set, prove every point has a connected neighborhood (so all connected components are open) and interp ...
Algebraic variety
In mathematics, algebraic varieties (also called varieties) are one of the central objects of study in algebraic geometry. Classically, an algebraic variety was defined to be the set of solutions of a system of polynomial equations, over the real or complex numbers. Modern definitions of an algebraic variety generalize this notion in several different ways, while attempting to preserve the geometric intuition behind the original definition.Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an ""algebraic variety"" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.The notion of variety is similar to that of manifold, the difference being that a variety may have singular points, while a manifold will not. In many languages, both varieties and manifolds are named by the same word.Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry among the other subareas of geometry.