finm314F06.pdf
... (c) Are these vectors are orthogonal in the inner product space R4 with the non-standard inner product hx, yi = x1 y1 + 2x2 y2 + 3x3 y3 + x4 y4 ? If not, use Gram-Schmidt to generate an orthogonal set from them. ...
... (c) Are these vectors are orthogonal in the inner product space R4 with the non-standard inner product hx, yi = x1 y1 + 2x2 y2 + 3x3 y3 + x4 y4 ? If not, use Gram-Schmidt to generate an orthogonal set from them. ...
TMA 4115 Matematikk 3 - Lecture 10 for MTFYMA
... Solving linear systems Given a linear system as x1 + 5x2 + 3x3 + 2x4 = 4 x1 − 2x3 + 2x4 = 0 2x2 + 4x3 + 2x4 = 1 find x1 , x2 , x3 , x4 which simultaneously satisfy (2). Use elementary operations to replace (2) with an equivalent system which is easier. But first: Rewrite (2) as an augmented matrix ...
... Solving linear systems Given a linear system as x1 + 5x2 + 3x3 + 2x4 = 4 x1 − 2x3 + 2x4 = 0 2x2 + 4x3 + 2x4 = 1 find x1 , x2 , x3 , x4 which simultaneously satisfy (2). Use elementary operations to replace (2) with an equivalent system which is easier. But first: Rewrite (2) as an augmented matrix ...
4 LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
... 2. Give an example to show that the union of two subspaces of a vector space V need not be a subspace of V. 3. Prove that any n + 1 vectors in Fn are linearly independent. 4. Define Kernel and Image of a homomorphism T. 5. Define an algebra over a field F. 6. What is eigen value and eigen vector? 7. ...
... 2. Give an example to show that the union of two subspaces of a vector space V need not be a subspace of V. 3. Prove that any n + 1 vectors in Fn are linearly independent. 4. Define Kernel and Image of a homomorphism T. 5. Define an algebra over a field F. 6. What is eigen value and eigen vector? 7. ...
Question 1 ......... Answer
... (a) The kernel of a matrix A is the set of all vectors ~x in the domain of A such that A~x = ~0. The image of A is the set of all vectors ~y in the target spaces of A such that there exists an ~x in the domain for which A~x = ~y. (b) For the kernel: If ~x1 , ~x2 ∈ ker(A), then A(~x1 + ~x2 ) = A~x1 + ...
... (a) The kernel of a matrix A is the set of all vectors ~x in the domain of A such that A~x = ~0. The image of A is the set of all vectors ~y in the target spaces of A such that there exists an ~x in the domain for which A~x = ~y. (b) For the kernel: If ~x1 , ~x2 ∈ ker(A), then A(~x1 + ~x2 ) = A~x1 + ...
Study Guide: Linear Differential Equations
... where f0 (x), f1 (x), . . . , fn (x) and g(x) are functions. For example, a second-order linear differential equation has the form f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x), and a third-order linear differential equation has the form f3 (x)y 000 + f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x). A linear diff ...
... where f0 (x), f1 (x), . . . , fn (x) and g(x) are functions. For example, a second-order linear differential equation has the form f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x), and a third-order linear differential equation has the form f3 (x)y 000 + f2 (x)y 00 + f1 (x)y 0 + f0 (x)y = g(x). A linear diff ...
Vector Spaces and Linear Maps
... Lemma 14.27. If L : V → W is a linear map then L(0) = 0 and L(−x) = −L(x) for all x∈V. Sums L + M and scalar multiples λL of linear maps are defined pointwise, just as are sums of functions. To add two linear maps, their domains and codomains must match. The composition L ◦ M of two linear maps is u ...
... Lemma 14.27. If L : V → W is a linear map then L(0) = 0 and L(−x) = −L(x) for all x∈V. Sums L + M and scalar multiples λL of linear maps are defined pointwise, just as are sums of functions. To add two linear maps, their domains and codomains must match. The composition L ◦ M of two linear maps is u ...
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.