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?
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
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?
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
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
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
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The Attempt at a Solution
Could someone explain what is meant by 'repeated...
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
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
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) 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
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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
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The Attempt at a Solution
a)I see that (M o L) takes all vectors in Rn and...