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If A is an nxn matrix with real eigenvalues, show that A=B+C where B is symmetric and C is nilpotent? (Hint: Use the fact that an nxn upper triangular matrix U with zeros on the main diagonal satisfies Uⁿ=0)

I know that:

A has real eigenvalues, therefore P^TAP is upper triangular

B is symmetric, orthogonally diagonizable, and has an orthonormal set of eigenvectors.

and because C is nilpotent, Cⁿ=0

I'm not sure how to solve this or how to use the hint because I'm not sure whether A has zeros on the main diagonal. I was hoping that I could take A-B=C and have A-B be an upper triangular matrix with zeros on the main diagonal but I don't think that is necessarily the case. How can I solve this and where does the hint come in?

Let A denote a symmetric nxn matrix.

a) If A=B² where B is symmetric, show that λ ≥ 0 for each eigenvalue of A.

b) If λ ≥ 0 for each eigenvalue of A, show that A=B² where B is symmetric.

For a I said that P⁻¹AP=P⁻¹B²P but I don't know how this can prove that the eigenvalues will be positive. I can't find any reference in my textbook as too what happens to the eigenvalues when you square a symmetric matrix. Any ideas?

a) compute p=proj∪(x).

b) Write x as the sum of a vector in U and a vector in U⊥.

c)Find the vector in U that is closest to x.

d) Find dim(U⊥)

I appreciate any help/insight that you can give. There is going to be another question like this on my next quiz and I have no idea how to do this.

a) If P is orthogonal, show that ||Px||=||x|| for every column x in Rⁿ.

b) If ||Px||=||x|| for every column x in Rⁿ, show that P is orthogonal.

(For b, I'm supposed to replace x by x+y but I don't understand how that works.)

If you could please give as much detail is possible. I need to learn how to do other questions like this. Thank you!

1. Let V and W be finite dimensional vector spaces. Show that:

a. If dim(V) ≤ dim(W), there exists a one-to-one linear transformation V→W.

b. If a one-to-one linear transformation V→W exists, then dimV ≤ dimW

I wrote:

Let T : V-->W

The dimension theorem gives that dimV=nullity(T) + rank (T)

and rank(T)≤dimW because Im(T) is a subspace of W.

Therefore, dimV - dimW ≤ nullity(T)

If dimV is less than or equal to dimW then, the nullspace is trivial and T must be one-to-one.

I'm not sure whether this is correct and I don't know how to prove the converse. I would appreciate any help that you can offer.

Let V=M_mn denote the space of all nxn matrices over lR. Let T : M_mn → lR denote the trace, that is T(A) = tr(A) for all matrices A∈M_mn. We know that T is a linear transformation with the property that T(AB)=T(BA) for all A, B∈M_mn. If S : M_mn → lR satisfies S(AB)=S(BA) for all A,B∈M_mn, show that S=aT for some a. (here aT is defined by (aT)(A)=aT(A) for all A∈M_mn).

[Hint: Let E_ij∈M_mn have (i,j)-entry 1 and all other entries 0. you may use the fact that these matrices multiply according to the formula E_ijE_kl={E_il if j=k, 0 if j≠k}. Show that S(E_ij)=0 if i≠j, and S(E_ii)=E_11 for all i.]

I have no idea how to do this proof. Could you please help?

Let V be a vector space. If v∈V define T_v : lR→V by T_v(x)=xv for all x in lR

a. Show that T_v : R→V is a linear transformation for each v in V.

b. Show that every linear transformation T: R→V arises as in (a) for a uniquely determined vector v in V.

I think this proof is going to be on my next quiz so I really need to understand how to solve it. Please help!?

2. Suppose T: U→V and S: V→W are linear transformations.

a. If ST is one-to-one and T is onto, show that S is one-to-one

b. If ST is onto and S is one-to-one, show that T is onto

I don't even know where to start with these. Any help you can give would be greatly appreciated!!!

Proof: Show that columns x and y are orthogonal in Rn if and only if ||x+y|| = ||x-y||

Answer:

Let x=(x1, x2,...,xn) and y = (y1, y2,...,yn).

columns x and y are orthogonal in Rn when x(dot product)y = 0

when x(dot)y=0, ||x+y|| = ||x||+||y|| and ||x-y|| = ||x||+||-y||=||x||+||y||

Therefore, when x and y are orthogonal in Rn, ||x+y||=||x-y||

Have I explained everything I need to? Is there a way to prove that ||x+y||=||x||+||y|| and ||x-y||=||x||+||-y|| when x(dot)y=0?

Find the best approximation to a solution of the following system:?

{2x-y=1}

{x+3y=0}

{3x+2y=2}

I reduced the matrix to:

{x + 0y = 3/7}

{0x + y = -1/7}

{0x + 0y = 1}

I just don't know how to find the best approximation. In my text book it says to calculate ||e||=||b-Ax|| where e is the error value but I'm not sure what values to plug in to get the correct answer.

Let A be nxn and nilpotent, that is A^m = 0 for some m>/=1.

a)if k is an eigenvalue of A, show that k=0.

b)Show that the characteristic polynomial c_A(x) of A is given by c_A(x) = x^n.

If rank A=n, show that {A^TC1, A^TC2,...,A^TCn} is a basis of R^n

Let A and B be square matrices of the same size. If A is similar to B and A^3-5A+2I = 0, show that B^3-5B+2I = 0.

Please give as much detail as you can. I think this may be on a test so I really need to understand how you solve the proof.

I got -t^2+5t-58/12 but thats not right. Do you know what I did wrong?

I got (5-36t)e^(4t) but that's not right. Do you know what I did wrong?

satisfying the initial condition y(1)=3^(1/3).

Please show how the answer was solved so that I can figure out how to solve similar problems. Also, What type of differential equation is this? (1st order, 2nd order, ect)

Find the particular solution of the differential equation

xy(prime)+y=6x^2+8x+2

satisfying the initial condition y(1)=−6

I have next to no idea how you to this so can you please explain how you get the answer so that I can apply it to similar questions? I would also like to know what type of differential equation this is (1st order, 2nd order, ect.) Thank You!