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Linear Transformation

Checking for a linear transformation

This chapter starts as follows rotations about the origin and all reflections in lines through the origin can be expressed as functions with rules of the form x ---> Ax where A is a 2 x 2 matrix any function with such a rule is called a linear transformation a linear transformation of the plane is a function of the form

Linear transformation

f(2,1) = (2,1) Either explain why f is not linear or write down the matrix that represents f . iI general how do you solve problems of this type?

Geometric Interpretation of the Subspace

Show that the set of all elements of R^2 of the form (a, -a), where a is any real number, is a subspace of R^2. Give a geometric interpretation of the subspace.

Groups : Isomorphism and Homomorphism

Note: S4 means symmetric group of degree 4 A4 means alternating group of degree 4 e is the identity Is there a group homomorphism $:S4 -> A4, with kernel $ = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}?


Show that innG is the normal subgroup of autG for any group G Note: innG = inner automorphism group of G aut G = automorphism group of G


If G = <X> and $:G->G1 is an onto homomorphism, show that G1 = <$(X)>, where $(X) = the set of $(x) given that x belongs to X.


Note: ~~ means an isomorphism exists. Moreover,if an isomorphism existed from G to G1 I would say G ~~ G1 Questions: If G is an infinite cyclic group, show that G ~~ Z (Z is the set of integers)

Powells Search method--Request 101767 if available

The attached file has some slides provided by my professor on the univariate method and powell's method. I am having trouble understanding, so I tried to work an example, but I am not getting very far. As you work the example, could you explain each step as you go. My professor tried but he and I both ended up frustrated, a

Solving Matrix Representation: Linear Transformation

** Please see the attached file for full problem description ** Let T be a linear operator on P_3 defined as follows: T(ax^3 + bx^2 + cx + d) = (a - b)x^2 + (c - d)x + (a + b - c). The matrix [T]_G which represents T with respect to the basis G which = {1 + x, 1 - x, 1 - x^2, 1 - x^3}. Show that

Kernel of Homomorphism

Please see the attached file for the fully formatted problems. Suppose &#61546; is an onto homomorphism from &#8484;16 to a group G of order 4. Find ker(&#61546;). Explain your answer.

Linear Algebra -- Linear Transformations

Let a be a fixed vector in R2. A mapping of the form L(x) = x+a is called a translation. Show that if a does not equal 0, then L is not a linear transformation. Describe or illustrate geometrically the effect of the translation. Thanks for your help!

Matrix Theory

See attached file for full problem description with symbols and equations. --- Definition 11.1 An orthogonal projection operator is a linear transformation such that and . Question: If W is a subspace of V, prove that P_w is an orthogonal projection. (P_w is P sub w)