THE ROTATION OF A COORDINATE SYSTEM AS A LINEAR
... That is, for any v ∈ R2,1 we have Tφ X v = X Av, or else, for the representation of the linear transformation Tφ we have RX X Tφ = Aφ . Now, if we rotate the vector ~a ∈ E2 by and angle β to form Tβ~a, and then we rotate it by the angle α to obtain the vector Tα Tβ~a, we can also arrive at the same ...
... That is, for any v ∈ R2,1 we have Tφ X v = X Av, or else, for the representation of the linear transformation Tφ we have RX X Tφ = Aφ . Now, if we rotate the vector ~a ∈ E2 by and angle β to form Tβ~a, and then we rotate it by the angle α to obtain the vector Tα Tβ~a, we can also arrive at the same ...
A 3 Holt Algebra 2 4-2
... 4-2 Multiplying Matrices A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. The multiplicative identity matrix is any square matrix, named with t ...
... 4-2 Multiplying Matrices A square matrix is any matrix that has the same number of rows as columns; it is an n × n matrix. The main diagonal of a square matrix is the diagonal from the upper left corner to the lower right corner. The multiplicative identity matrix is any square matrix, named with t ...
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WHEN IS F[x,y] - American Mathematical Society
... Proposition 3. Let f be a linear polynomial in R = F[x, y] that is not associated to a central polynomial. Then ff is a C-atom. Proof. Let f = ax + by + c. If either a or b is 0 then / G F[y] or F[x], respectively, so ff is a C-atom by Proposition 2. Suppose now that both a and b are nonzero and // ...
... Proposition 3. Let f be a linear polynomial in R = F[x, y] that is not associated to a central polynomial. Then ff is a C-atom. Proof. Let f = ax + by + c. If either a or b is 0 then / G F[y] or F[x], respectively, so ff is a C-atom by Proposition 2. Suppose now that both a and b are nonzero and // ...
Condensation Method for Evaluating Determinants
... its determinant, det A or ⎜A ⎜, is equal to ad–bc. Given a matrix A, its determinant provides useful geometric and algebraic information about the matrix. Geometrically, the row entries of an n × n matrix A define the edges of a parallelepiped in ndimensional space, of which the volume is simply the ...
... its determinant, det A or ⎜A ⎜, is equal to ad–bc. Given a matrix A, its determinant provides useful geometric and algebraic information about the matrix. Geometrically, the row entries of an n × n matrix A define the edges of a parallelepiped in ndimensional space, of which the volume is simply the ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.