(a) Give an example of a nonempty subset U in R2 such that U is closed under multiplica- tion by scalars, but U is not a linear subspace of R2. (b) Give an example of a nonempty subset W in R2 such that W is closed under addition, but W is not a linear subspace of R2.
Q8 6 Points Let T : R2 + Rº be a linear transformation with PT(x) = x2 – 1. Decide whether or not such a T is always diagonalizable. Justify your answer.. Q8.2 3 Points Determine/Compute the linear transformation T2 : R2 + R2, VH T(T(u)).
Let T : R3 → R2 be a linear map. Recall that the image of T, Im(T), is the set {T(i) : R*) (a) Suppose that T(v)- Av. Describe the image of T in terms of A Using this description, explain why Im(T) is a subspace of R2. (b) What are the possible dimensions of Im(T)? (c) Pick one of the possible dimensions and construct a specific map T so that Im(T) has that dimension.
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be
the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l
A^T=-A}.
(a) Show that W is a subspace of M2x2(R)
(b) Find a basis for W and determine dim(W).
(c) Suppose T: M2x2(R) is a linear transformation given by
T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You
do not need to verify that T is linear.
3. (17 points)...
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Consider a linear transformation F : R2→R2
In lectures it is shown that the reflection in a subspace can be calcu- lated by Rw(u) = 2 prw(u) – u. Use this formula to find the standard matrix of the linear transformation described above. and hence deter- mine the image of the reflection of the y-axis in the line y = 2x.
a. Let be an differential operator. Show that L is a linear operator. b. Let be an differential operator. Show that the kernel of L is a vector space c. Let . Show that the set of functions which satisfy L(u) = g(x,t) form an affine linear subspace. L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2) L=(a252 -2)
Suppose T: ℝ3→ℝ2 is a linear transformation. Let U and V be the
vectors given below, and suppose that T(U) and T(V) are as given.
Find T(3U+3V).
Suppose T: R->R2 is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(3U+3V). 5 5 6 T(V) 6 =n 2 -3 T(U) V = 3 -4 3 -4
Suppose T: R->R2 is a linear transformation. Let U and V...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
2. Let T: P2 + R2 be the linear transformation given by (a-6) T(a + bx + cx?) = | 16+c) Find ker T and im T.