B1) This question concerns the following two subsets of :

(a) Show that , and find a vector in that does not belong to T. [3]
(b) Show that T is a subspace of . [4]
(c) Show that S is a basis for T, and write down the dimension of T. [7]
(d) Find an orthogonal basis for T that contains the vector . [3]
(e) Express the vector of T as a linear combination of the vectors in your orthogonal basis for T. [3]

B2 This question concerns the function t given by the rule

(a) Use the strategy in Unit 4, section 1, to show that t is a linear transformation. [3]
(b) Write down the matrix for t with respect to the standard basis in both the domain and codomain. [1]
(c) Determine the matrix of t with respect to the domain basis S of and the standard basis in the codomain. [2]
(d) Determine the matrix of t with respect to the domain basis S of basis and codomain basis of U. [5]
(e) Find the kernel of t, and states its dimension. [3]
(f) Let be a linear transformation given by . Determine the matrix of m and with respect to the standard basis in both the domain and the codomain. [6]

Please see the attached file for the fully formatted problems.

Linear transformations and subspaces are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

(7) Let F=R, let V=R^2 and let T be the linear transformation from V to V which is rotation clockwise about the origin by pi-radians. Show that every subspace of V is an
F[X]-submodule for this T.
Here F[X] is a polynomial domain where the coefficient ring is a field F.

Please use a step-by-step method to solve the two following exercises:
Exercise 1:
Consider the linear system x' = Ax, where A is a 2 x 2 matrix with lamda in the diagonal as follows;
A = [lamda, -2]
[1 , lamda],
and lamda is real. Determine if the system has a saddle, node, focus, or center at the ori

Determine whether the following are lineartransformations from C[0,1] into R^1.
L(f) = |f(0)|
L(f) = [f(0) + f(1)]/2
L(f) = {integral from 0 to 1 of [f(x)]^2 dx}^(1/2)
Thanks so much. :)

Please help. I always have a hard time with Linear Algebra. What's the difference between mapping from R3 into R2 and mapping from R2 into R3?
Why is the following not a linear transformation from R3 into R2?
L(x) = (1 + x1, x2)^T
And why is this one not a linear transformation from R2 into R3?
L(x) = (x1, x2, 1)^T
Th

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)

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)

How to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1