Exercise 2.5.1 Suppose T : R n ? R n is a linear transformation.
Prove that...
Exercise 2.5.1 Suppose T : R n ? R n is a linear transformation.
Prove that T is an isometry if and only if T(v) · T(w) = v · w.
Recall that an isometry is a bijection that preserves distance.
. Let T : R n → R m be a linear transformation and A the
standard matrix of T. (a) Let BN = {~v1, . . . , ~vr} be a basis
for ker(T) (i.e. Null(A)). We extend BN to a basis for R n and
denote it by B = {~v1, . . . , ~vr, ~ur+1, . . . , ~un}. Show the
set BR = {T( r~u +1), . . . , T( ~un)} is a...
Prove that any linear transformation ? : R? → R? maps a line
passing through the origin to either the zero vector or a line
passing through the origin. Generalize this for planes and
hyperplanes. What are the images of these under linear
transformations?
Prove that any linear transformation ? : R? → R? maps a line
passing through the origin to either the zero vector or a line
passing through the origin. Generalize this for planes and
hyperplanes. What are the images of these under linear
transformations?
For problems 1-4 the linear transformation T/; R^n -
R^m is defined by T(v)=AV with
A=[-1 -2 -1 -1
2 1 0 -4
4 6 1 15]
1. Find the dimensions of R^n and R^m (3 pts)
2. Find T(<-2, 1, 4, -1>) (5 pts)
3. Find the preimage of <-6, 12, 4> (8
pts)
4. Find the Ker(T) (8 pts)
Consider the linear transformation T : P1 → R^3 given by T(ax +
b) = [a+b a−b 2a]
a) find the null space of T and a basis for it
(b) Is T one-to-one? Explain
(c) Determine if w = [−1 4 −6] is in the range of T
(d) Find a basis for the range of T and its dimension
(e) Is T onto? Explain
Consider the linear transformation T: R^4 to R^3 defined by T(x,
y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w)
a) Find the dimension and basis for Im T (the image of T)
b) Find the dimension and basis for Ker ( the Kernel of T)
c) Does the vector v= (2,3,5) belong to Im T? Justify the
answer.
d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the
answer.
(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear
transformation such that • (0, 0, 1, 0) and (0, 0, 0, 1) lie in the
kernel of T, and • all vectors of the form (x1, x2, 0, 0) are
reflected about the line 2x1 − x2 = 0.
(a) Compute all the eigenvalues of A and a basis of each
eigenspace.
(b) Is A invertible? Explain.
(c) Is A diagonalizable? If yes,...
Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) =
(2,3,-5) and T( e⃗2 ) = (-1,0,1).
Determine the standard matrix of T.
Calculate T( ⃗u ), the image of ⃗u=(4,2) under T.
Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the
image of ⃗w=2⃗u−v⃗ .
4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.
Consider the linear transformation T: R2x2 ->
R2x2 defined by T(A) = AT - A.
Determine the eigenvalues of this linear transformation and
their algebraic and geometric multiplicities.