Linear Transformations : Basis, Kernel, Image, Onto, One-to-one and Matrices

1) Define a linear transformation....
a) Find a basis for Ker T.
b) Find a basis for Im T.
c) Is T an onto map?
d) Is T a one-to-one map?

2) Define a linear transformation...
a) Find the matrix for T with respect to the standard basis.
b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and the standard basis for R2.

Please see the attached file for the fully formatted problems.

Ker T is the set of all elements that are mapped into zero.
So a + b = 0 and b + c = 0
So b = -a and c = -b = a
Ker T = { ax^2 - ax + a where a is any real number}

I don't know what format you are using for basis; if you treat the coefficients as a vector, the basis is a single vector (1,-1,1) Only one ...

Solution Summary

Linear Transformations, Basis, Kernel, Image, Onto, One-to-one and Matrices are investigated.

See attached file.
Consider the relationship between the matrices A and B, explore the relationship between A and B andmatrices representing other lineartransformations derived from F and G. Explore 2 examples for example F+G for various and , or the composite function FG/
Check th

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)

2. Let T and S be matrix multiplication transformations from R^2 into R^3, described as
T[x, y] = [1,2; 1,1][x,y] and S[x,y] = [7,4; 6,7][x,y]
Find the transformations 2T - S, ST and TS. Do T and S commute?
3. Let U = {all[a,c; 5a,3c]},
that is U is the set of all 2 x 2 matrices A such that a(12) = 5a(11), a(22) = 3a(23)

A linear transformation L:V->V is said to be idempotent if L dot L = L. If L is idempotent, show that there exists a basis S={a1,a2,...,an} for V such that L(ai)=ai for i= 1,2,...,r and L(aj) = 0v for j= r+1,...,n, where r= p(L). Describe the matrix representing L with respect to the basis S.

1) Let u=(2,3,0), and v=(-1,2,-2). Find
a) ||u + v||
b) ||u|| + || v ||
c) Find two vectors in R³ with norm 1 orthogonal to be both u and v
d) Find norm of vector u / || u ||
2) For which values of t are vectors u = (6, 5, t), and v = (1,t,t) orthogonal?
3)
a) Find the standard matrix [T] for the linear tra

Are the following examples lineartransformations from p3 to p4? If yes, compute the matrix of transformation in the standard basis of P3 {1,x,x^2} and P4 {1,x,x^2,x^3}.
(a) L(p(x))=x^3*p''(x)+x^2p'(x)-x*p(x)
(b) L(p(x))=x^2*p''(x)+p(x)p''(x)
(c) L(p(x))=x^3*p(1)+x*p(0)

Please refer to attachment for proper formatting.
(i) Write down the matrix, A, that represents a shear with x-axis invariant in which the image of the point (1,1) is (4,1).
(ii) Given matrix B (in the attachment), describe fully the geometrical transformation represented by B.
(iii) Given matrix C (in the attachment),

See the attached file.
1. Find a basis B of R^n such that the B-matrix B of the given linear transformation T is diagonal.
Reflection T about the line in R^n spanned by [2;3
2. Which of the subsets of P_2 given below are subspaces of P_2? Find the basis for those that are subspaces.
i) {p(t):p(0)=2}
ii) {p(t):p(2)=0}
i