Question

Let T be the function from R2 to R3 defined by T ( (x,y) ) = (x, y, 0). Prove that T is a linear transformation, that it is 1-1, but that it is not onto.

Answer #1

Does there exist a linear transformation T from R2 to
R3 that is onto but not 1-1? Support your answer with a
proof or counterexample, as appropriate.

10 Linear Transformations. Let V = R2 and W = R3. Define T: V →
W by T(x1, x2) = (x1 − x2, x1, x2). Find the matrix representation
of T using the standard bases in both V and W
11 Let T :R3 →R3 be a linear transformation such that T(1, 0, 0)
= (2, 4, −1), T(0, 1, 0) = (1, 3, −2), T(0, 0, 1) = (0, −2, 2).
Compute T(−2, 4, −1).

(a) Let T be any linear transformation from R2 to
R2 and v be any vector in R2 such that T(2v)
= T(3v) = 0. Determine whether the following is true or false, and
explain why: (i) v = 0, (ii) T(v) = 0.
(b) Find the matrix associated to the geometric transformation
on R2 that first reflects over the y-axis and then
contracts in the y-direction by a factor of 1/3 and expands in the
x direction by a...

Let T be the linear transformation from R2 to R2, that rotates a
vector clockwise by 60◦ about the origin, then reﬂects it about the
line y = x, and then reﬂects it about the x-axis.
a) Find the standard matrix of the linear transformation T.
b) Determine if the transformation T is invertible. Give detailed
explanation. If T is invertible, ﬁnd the standard matrix of the
inverse transformation T−1.
Please show all steps clearly so I can follow your...

b) More generally, find the matrix of the linear transformation
T : R3 → R3 which is u1
orthogonal projection onto the line spanu2. Find the matrix of T.
Prove that u3
T ◦ T = T and prove that T is not invertible.

Let T:V-->V be a linear transformation and let T^3(x)=0 for
all x in V. Prove that R(T^2) is a subset of N(T).

Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m
R^n is a linear transformation

Let U = Z+ ∪ { 0 }.
Let R1 = { ( x, y ) | x ≠ y }
Let R2 = { ( x, y ) | x = y }
Let R3 = { ( x, y ) | x ≥ y }
Let R4 = { ( x, y ) | x ≤ y }
Let R5 = { ( x, y ) | x > y }
Let R6 = { ( x, y...

Consider the linear transformation P : R3 → R3 given by
orthogonal projection onto the plane 3x − y − 2z = 0, using the dot
product on R3 as inner product.
Describe the eigenspaces and
eigenvalues of P, giving specific
reasons for your answers. (Hint: you do not need
to find a matrix representing the transformation.)

Let the function f on R2 be f(x,y) =
x3−3αxy+y3,∀(x,y) ∈R2, where α ∈R
is a parameter.
Show that f has no global minimizer or global maximizer for any
α.

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