If matrix A is similar to #, is T{M) = AM MB an isomorphism from R2x2 to R2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M + h from R2*2 to M2*2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = IM from R2x2 to R2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = (sum of the diagonal entries of M) from M2x2 to R

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{M) = det(Af) from R2x2 to R

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M2 fromIR2x2 toIR2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. 7(Af) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M from IR2x2 to M2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = S~'MS. where S = R2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{M) = PMP~l. where P =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = PMQ, where P = to] and Q = 3 5 7 11 , from P2x2 - i2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. Tic) = c 2 3 4 5 from R to R2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{M) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{M) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = 2 3 5 7 2 0 0 3 M-M M-M 2 3 5 7 4 0 0 5

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(M) = M '2 o' 3 o' 0 3 0 4 M from R2 x2 to R2x2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(x + iy

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(x -h iy

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T (x + iy

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(x + ix

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(x + iy

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{f(t))

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{f(0)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T ( / U))

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(fU))

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{f(D)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{f(0)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{fU))

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f(t))

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T{.fU)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f()

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(m)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. In Exercises 33 through 36, V denotes the space of infinite sequences of real numbers. T(x 0 , x\, x2. *3, * 4 , . . .) = 0*o, A*2, *4, ) from V to V (we are dropping every other term)

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. In Exercises 33 through 36, V denotes the space of infinite sequences of real numbers. T(x0, x i, . ) = (0, *0, *1, *2* ) from V to

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. In Exercises 33 through 36, V denotes the space of infinite sequences of real numbers. 7"(/(/)) = (/(0 ),/'(0 ),/"(0 ),/"'(0 ),...) from /> to V, where P denotes the space of all polynomials

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. In Exercises 33 through 36, V denotes the space of infinite sequences of real numbers. T(f(0) = (/(0), /(l),/(2),/(3)....)from/5toV. where P denotes the space of all polynomials

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f) = / + / ' from C00 to C *

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r ( /) = / + /" from C to C

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f) = /" - 5 /' + 6 / from C to C00

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T (/) = f" + 2 f + f from Cx to Cx

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r(/(f)) = fit) + /"(/) + sin(r) from C to C

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r(/(/>) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f(0) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. 7" (/(/)) =

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = t{f(D) from P to P

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r(/(/)) = (/ l)/(f) from P to

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r(/(/)) = Jj /U ) from P to P

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(/(/)) = /'(f) from P to

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. T(/(r)) = /(r2) from P to P

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms. r(/(/)) = J(t + 2^~/(/) from P t0 P

Find kernel and nullity of the transformation in Exercise 13.

Find kernel and nullity of the transformation in Exercise 6.

Find image, rank, kernel, and nullity of the transformation in Exercise 25.

Find image, rank, kernel, and nullity of the transformation in Exercise 22.

Find image and kernel of the transformation in Exercise 33.

Find image, rank, kernel, and nullity of the transformation in Exercise 30.

Find kernel and nullity of the transformation in Exercise 39.

Find image and kernel of the transformation in Exercise 34.

Find image, rank, kernel, and nullity of the transformation in Exercise 23.

Find image, rank, kernel, and nullity of the transformation in Exercise 42.

Find image and kernel of the transformation in Exercise 45.

Find image and kernel of the transformation in Exercise 48.

Define an isomorphism from P3 to R3, if you can.

Define an isomorphism from P3 to R2x2, if you can.

We will define a transformation T from R',xm to F(Rn\ R n)\ recall that F(R"',R") is the space of all functions from Rm to Rn. For a matrix A in R',xm, the value T(A) will be a function from Rm to Rn\ thus we need to define (T(A))(v) fora vector t) in Rm. We let (7'(A))(i5) = Av. a. Show that T is a linear transformation. b. Find the kernel of T. c. Show that the image of T is the space L(Rm, R) of all linear transformation from Rm to Rn. (See Exercise 4.1.49.) d. Find the dimension of L(Rm, R").

Find the kernel and nullity of the linear transformation T(f) = f - f f from C to C.

For which constants k is the linear transformation T(M) = 2 3 0 4 M - M 3 0 0 k an isomorphism from ] 2x2 to!

For which constants k is the linear transformation = ]-[J an isomorphism from R2x2 to R2x2?

If matrix A is similar to #, is T{M) = AM MB an isomorphism from R2x2 to R2x2?

For which real numbers to, c 1........cn is the linear transformation ./(co) /U l) f(Cn) 1 + I 9

Does there exist a polynomial f(t) of degree < 4 such that /(2) = 3, /(3) = 5, /(5) = 7, /(7) = 11, and /(ll) = 2? If so, how many such polynomials are there? Hint: Use Exercise 70.

In Exercises 72 through 74, let Zn be the set of all polynomials of degree < n such that f (0) = 0. Show that Zn is a subspace of Pn, and find the dimension of Z.

In Exercises 72 through 74, let Zn be the set of all polynomials of degree < n such that f (0) = 0. Is the linear transformation 7 (/(f)) = Jq / (jc) dx an isomorphism from Pn~\ to Znl

In Exercises 72 through 74, let Zn be the set of all polynomials of degree < n such that f (0) = 0. Define an isomorphism from Zn to Pn~\ (think calculus!).

Show that if 0 is the neutral element of a linear space then 0 + 0 = 0 and kO = 0, for all scalars k.

Show that if 7 is a linear transformation from V to W, then T(0y) = Oyy, where 0y and 0w are the neutral elements of V and W, respectively.

If 7 is a linear transformation from V to W and L is a linear transformation from W to U, is the composite transformation LoT from V to U linear? How can you tell? If T and L are isomorphisms, is L o T an isomorphism as well?

Let R+ be the set of positive real numbers. On R+ we define the exotic operations jc y = xy (usual multiplication) and k O jc = xk. a. Show that R+ with these operations is a linear space; find a basis of this space. b. Show that T(x) = ln(jc) is a linear transformation from R+ to R, where M is endowed with the ordinary operations. Is T an isomorphism?

Is it possible to define exotic operations on R2, so that dim(R2) = 1?

Let X be the set of all students in your linear algebra class. Can you define operations on X that make X into a real linear space? Explain.

In this exercise, we will outline a proof of the rank-nullity theorem: If T is a linear transformation from V to W, where V is finite dimensional, then dim( V) = dim(im T) + dim(ker T) = rank(7) + nullity(T). a. Explain why ker(7) and image (7) are finite dimensional. Hint: Use Exercises 4.1.54 and 4.1.57. Now consider a basis v\,..., vn of ker(7), where n = nullity(7), and a basis w\,...,wr of im(7), where r = rank(7). Consider elements u\.......ur in V such that 7(m;) = u)j for i = 1,..., r. Our goal is to show that the r + n elements u i,..., ur, i>i,..., vn form a basis of V; this will prove our claim. b. Show that the elements u i,..., ur, i>i,..., vn are linearly independent. Hint: Consider a relation ci u i + + crur + d\ v\ + + dn vn = 0, apply transformation T to both sides, and take it from there. c. Show that the elements u\,..., wr, U|____ vn span V. [Hint: Consider an arbitrary element v in V, and write T(v) = d\ w\ + + drwr . Now show that the element v d \u \-------- drur is in the kernel of 7, so that v d\u\ ... drur can be written as a linear combination of uj.......

Prove the following variant of the rank-nullity theorem: If T is a linear transformation from V to Wy and if ker T and im T are both finite dimensional, then V is finite dimensional as well, and dim V = dim(ker T) + dim(im T).

Consider linear transformations T from V to W and L from W to U. If ker T and ker L are both finite dimensional, show that ker(L o T) is finite dimensional as well, and dim (ker(LoD) < dim(ker 7)+ dim(kerL). [Hint: Restrict 7 to ker(L o 7) and apply the rank-nullity theorem, as presented in Exercise 82.]

Consider linear transformations 7 from V to W and L from W to U. If ker 7 and kerL are bothj finite dimensional, and if im 7 = W, show that ker(L o 7) is finite dimensional as well and tha^ dim (ker(L o7)) = dim(ker 7) + dim(ker L).