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Homework Statement Let f(x, y) = x^2y^3 + xy. Is there a direction at (-1; 2) in which the rate of change of f is equal to 18? Justify your answer. Homework Equations The Attempt at a Solution plugging this into the directional derivative formula, i get 18 = -v1 + 13v2, where...

Homework Statement Is The space P2 is isomorphic to the space of all 3 × 3 diagonal matrices. Homework Equations The Attempt at a Solution I know that P2 is isomorphic to vectors with 3 components so i think this statement is true, is it?

A = [a b; c d] det (A) = ad-bc A transpose = [a c; b d] det A transpose = ad - bc = det A

lol that's funny. What i meant is May you show me such a proof, please?

ohhh it works o.o can i see a rigorous proof of this?

Homework Statement I don't understand why the determinant of a matrix is equal to its transpose...how is this possible? Homework Equations The Attempt at a Solution

ohhhhh is it m - n?

Homework Statement Suppose that dim V = m and dim W = n with M>=n . If the linear map A : V -> W is onto, what is the dimension of its kernel? Homework Equations The Attempt at a Solution Onto, means that every vector in W has at least one pre-image therefore, the kernel can...

Homework Statement Six digits from the numbers 2, 3, 4, 5, 6, 7, 8 are chosen and arranged in a row without replacement. Find the probability that the digits 2 and 3 appear in the proper order but not consecutively Homework Equations The Attempt at a Solution i know that the...

I just tried several similar matrices but they all share the same eigenvector O.o Can i get an example where two similar matrices have different eigenvectors?

ohhh i see... since they can have the same eigenvalues, does this mean that the matrices can also have the same eigenvectors?

oh wait...B = P^-1 AP ...so what i said is wrong... how can i manipulate A P^-1 P to look like P^-1 AP?

Av = lambda v (AP^-1 P)v = lambda v Bv = lambda v i think?

what do u mean by the same characteristic equation?

Homework Statement Let A and B be similar matrices a)Prove that A and B have the same eigenvalues Homework Equations None The Attempt at a Solution Firstly, i don't see how this can even be possible unless the matrices are exactly the same :S

u mean like: (Av +Bv) = lambda*v + mui*v (A+B)v = (lambda + mui) v

Homework Statement Suppose that v is an eigenvector of both A and B with corresponding eigenvalues lambda and mui respectively. Show that v is an eigenvector of A+B and of AB and determine the corresponding eigenvalues Homework Equations The Attempt at a Solution Av = lambda*v Bv...

ohhhhhh i think i got it now det(A-lambda*I) =(lambda-lambda_1)(lambda-lambda_2)...(lambda-lambda_n) if lambda =0, then we have det(A) =(lambda_1)(lambda_2)...(lambda_n) but, can we just set lambda = 0 like that?

the factors of polynomials are the roots of the polynomials i think... det(A)=(lambda-lambda_1)(lambda-lambda_2)...(lambda-lambda_n) so the eigenvalues are lambda_1...lambda_n

How can i write our the characteristic polynomial if we're dealing with a general nxn matrix?

to find p(0), i wud just sub 0 in the place of every variable and solve. I will be left with a constant, if there is a one Is this wat u r asking? but how does this relate to the question? =S

Homework Statement Let A be an nxn matrix, and suppose A has n real eigenvalues lambda_1, ...lambda_n repeated according to multiplicities. Prove that det A = lambda_1...lambda_n Homework Equations None The Attempt at a Solution Could someone explain what is meant by 'repeated...

Is there another way to prove this question other than finding the determinant for the matrices?

ohh sorry. so, um, are u asking me to apply that formula to the determinants in the question and show that the determinants are the same?

um actually, that's wat i meant o.o

To calculate the determinant of a 3x3 matrix, u take the sum of the cofactors multiplied by the number in each entry of the original matrix . which is then also multiplied by (-1)^n, where n represents the position of the entry

A determinant is a function depending on n that associates a scalar, det(A), to an n×n square matrix A. I don't see how this helps me tho.. I'm sorry T_T

Homework Statement Prove det [a+p b+r c+s; d e f; g h i] = det [ a b c; d e f; g h i] + det [p r s; d e f; g h i] Homework Equations none The Attempt at a Solution i'm not sure how to prove this though its seems obviously true =S

i can't think of a counterexample such that a matrix P and a matrix A whose P^-1AP is not a diagonal matrix. What is the counterexample that u know?

hmm so can i say : det(P-1 AP) = det(P^-1)det(A)det(P) = 1/det(P) det(A) det(P) =det(A) =S, for some reason, i don't think this proof is rigorous enough

Homework Statement Let P be an invertible nxn matrix. Prove that det(A) = det(P^-1 AP) Homework Equations none The Attempt at a Solution P^-1 AP gives me a diagonal matrix so to find the determinant , i just multiply the entry in the diagonal. However, i don't understand why P^-1...

To find the coordinates of a vector with respect to a basis, row reduce the vector with the new basis. To:AUMathTutor I think D works out to be [0 1 0 0 ; 0 0 2 0; 0 0 0 3] if we include polynomials of degree 3 in P3

So v1 = L(u1) = 0 v2 = L(u2) = 1 v3 = L(u3) = x v4 = L(u4) = x^2 How am i suppose to rewrite this to represent the matrix D?...because this is now the standard basis for P2

um...then is it the set of different polynomials? {1, x, x^2, x^3}...i think

ohh wait wait , I meant [ 1 0 0 0; 0 1 0 0 ; 0 0 1 0; 0 0 0 1]

The standard basis is [1 0 0; 0 1 0 ; 0 0 1]

Homework Statement Let D : P3--> P2 be differentiation of polyonimals. Determine the matrix of D with respect to the standard basis of P3. Homework Equations None The Attempt at a Solution I think D=[1 0 0; 0 1 0; 0 0 0]. This is from inspection though because I know that the...

If A and B are isomorphic,then they are structurally identical. I just know this definition from class. What I get from this is that A and B have the same number elements and they are a one to one mapping :S

\phi, what exactly is this notation? and how do I show that a) is an isomorphism? Could you elaborate some more on b). I'm not quite sure i understand what you mean

a)There are n+1 a's?

a) we have n elements which we can choose in any way we like b)I am wondering how the rank of the matrix can possibly be 3 because in a 2x3 matrix, the highest rank it can have is 2...right?

a)Don't you mean to say dim(null(D)) + rank(D) = dim(P^{n}) Because P^{n-1} refers to the range space of D... The rank nullity-theorem states dim(null(D)) + rank(D) = dim(D) and in this case the dimension of D is Pn b) I think I see what you are tying to say. For the matrix A=[d e f; 0 0 0]...

Homework Statement Suppose that the span {v1,...,vn} = V and let L:V-->W be an onto linear mapping. Prove that span {L(v1),...,L(v2)} = W Homework Equations None The Attempt at a Solution I think for this question, we just have to show that if vi, where i is a real number, is a...

Homework Statement Suppose that {v1,...,vn} is a linearly independent set in a vector space V and let L:V --> W be a one-to-one linear mapping. Prove that {L(v1),...,L(vn)} is linearly independent. Homework Equations None The Attempt at a Solution If L is a one-to-one linear...

Homework Statement For each of the following pairs of vectors, define an explicit isomorphism to establish that the spaces are isomorphic. Prove that your map is an isomorphism. a)P3 and R4 b)P5 and M(2,3) Homework Equations None The Attempt at a Solution a)I know that P3 and R4...

a)All coefficients other than the coefficient of x^0 must be equal to 0 in order for the polynomial to have a derivative of 0. Therefore, n coefficients are constrained by the definition of the null space and we can choose one coefficient any way we like (i.e. the coefficient of x^0) Therefore...

a)Given an arbitrary element of Pn, which looks like, [tex]a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x^1 + a_0 x^0[\tex], how many of the coefficients are constrained by the definition of the null space of D? One of the coefficients is constrained by the definition of the nullspace of D. It...

a) The subspace of polynomials that have derivative equal to zero are those with x^0. So then this means that the rank is 1 and the nullity is 0 by the rank-nullity theorem? b) Applying L to a general matrix M = [a b c; d e f] gives [d e f; 0 0 0]. For M to be the nullspace of L, then d e and...

Homework Statement By considering the dimensions of the range or null space, determine the rank and the nullity of the following linear maps: a) D:Pn --> Pn-1, where D(x^k)=Kx^k-1 b) L:M(2,3) --> M(2,3) where L([a b c; d e f])=[d e f; 0 0 0] c) Tr:M(3,3) --> R, where Tr(A)=a11+a22+a33 (the...

Homework Statement Let L : Rn --> Rm and M : Rm --> Rp be linear mappings. a)Prove that rank( M o L) <= rank(L). b)Give an example such that the rank(M o L) < rank(M) and rank(L) Homework Equations None The Attempt at a Solution a)I see that (M o L) takes all vectors in Rn and...