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Hi I need some help with the following question:
For what real values x is the symmetric real matrix A=
congruent to the matrix
and
I have put it into quadratic form and attempted to reduce it but am not getting anywhere. I get the 2x2 identity matrix except the (1,1) entry is x. As for the second congruent matrix it can be written as 2xy in quadratic form which can be written as (x+y)^2 - x^2 - y^2. That is as far as I could get. If anyone could help me by tomorrow morning I'd really appreciate it
I just wanted to ask a couple of things regarding the first 2 statements:
Does span(B') consist of 8 non-zero elements and 10 zeros?
And for the second statement, span(0,0)=span(v1,v2,v3,v4) where v2=-v1 and v4=-v3 as you said. So it is false unless v1=v2=v3=v4=0 is what you are saying?
I would really appreciate it if you could clarify this for me, thanks!
Hi I need help with the following questions:
State whether these are true or false with justification:
1) Every 18-dimensional vector space V contains a 8-dimensional subspace W
2) For all vectors v1,v2,v3,v4 which are elements of R8, span(v1+v2,v3+v4)= span(v1,v2,v3,v4)
3) There exists a 5x5 matrix whose characteristic polynomial is (x-1)^5 and minimal polynomial is x-6
Thanks in advance!
Hi I need some help with proving if some statements are true or false with brief reasons needed.
1) If A is a square matrix whose entries lie in a field K then the determinant of A cubed is an element of K
2) For every field K and every c that is an element of K, there exists a 4x4 matrix A with entries in K such that det(A)= c
3)R^2 is a subspace of C^2( R is the set of real numbers and C the set of complex numbers)
4) The set of integer vectors Z^3 = {[ x y z] transposed: x,y,z are elements of Z} is a subspace of R^3
Thanks in advance:D
Hi I need some help on a few proofs.
1) Prove that if {v1,....vn} is an orthonormal basis for R^n then {Qv1,....Qvn} is an orthonormal basis for R^n where Q is an orthogonal square matrix
2) S is a subspace of R^n with an orthogonal basis {v1,.....,vp} and {w1,......,wq} is an orthogonal basis for the orthogonal complement of S.
a-> Explain why {v1,...,vp,w1,......wq} is an orthogonal set that spans R^n
b-> Use part a to show that dim(S)+dim(orthogonal complement of S)=n
Thanks alot in advance!
Hi I need help on a couple of questions.
1) v1,...vr are vectors in Rn and S=Span(v1,...vr). Show that x is an element of the orthogonal complement of S if and only if x is orthogonal to each vj for j=1....r
2)A is an arbitrary matrix, x is in the column space of A. If the A^Tx=0, what is x? (A^T meaning the transpose of A)
Thanks in advance!
Let u,v and w be vectors in a vector space. Prove that if u+v=w+v then u=w.
I think its something to do using the cancellation law along with the zero element and inverse axioms of vector spaces. Can someone help me clarify this, thanks!
Hi im stuck on this question, can anyone help please?
Let A and D be square nxn matrices. Show that if D is diagonal with
diagonal entries a1,....,an, that is dii = ai where D = (dij), then
a)AD is the matrix whose j-th column is aj times the j-th column of A
b)DA is the matrix whose i-th row is ai times the i-th row of A
thanks
thanks alot!
Hi i need help with this question:
Let G be the symmetric group S3.
a)What are the orders of elements of G?
b)Show that G contains just one subgroup of order 3 and three subgroups of order 2
Thanks in advance
thanks for the help!
Hi I'm stuck on 2 questions on algebra.
1)Let G be a group with 4 elements {e,a,b,c} in which e is the identity and a^2=b. Write down the Cayley table of G, explaining your working.
2)G is the set of all rotations about the origin of the real Euclidean plane, the operation is the composition of the mappings. Is this a group?Justify your answer
I'd appreciate any help, thanks
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