### Kernels, image, nullity and rank.

Find the matrix A of T with respect to the standard basis...of both V and W. Compute the kernel, the image, the nullity and the rank of T. (See attachment for full question)

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Find the matrix A of T with respect to the standard basis...of both V and W. Compute the kernel, the image, the nullity and the rank of T. (See attachment for full question)

Verify that the solution of u"=f(x), u(0)=0, u(1)=0 given by u(x)= the integration from 0 to 1 of k(x,y)f(y)dy. Use Leibniz rule. (See attachment for full question)

1. Let T be any automorphism of G, show that ZT<(subset) Z. If G is a group and Z is the center of G.

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

Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

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)}?

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)

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

** 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

Please see the attached file for the fully formatted problems. Suppose  is an onto homomorphism from ℤ16 to a group G of order 4. Find ker(). Explain your answer.

Extension of A_5

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!

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)

Please see attached file.

Please see attached file

Is the following transformation linear? T: R^3 -> R^2 defined by T(x,y,z)=(x,y)