SOLUTION: A^T denotes the transpose of a matrix A. Show that <A,B> = Tr(B^T,A) defines an inner product on R^(n x n) Let U be the set of symmetric 2 x 2 matrices with real entries. U is

Question 15127: A^T denotes the transpose of a matrix A.
Show that = Tr(B^T,A) defines an inner product on R^(n x n)
Let U be the set of symmetric 2 x 2 matrices with real entries. U is subspace of R (2x2). Find an orthonormal basis of U with respect to the above inner product.
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
A : nxn square matrix over R. (missing)

A^T denotes the transpose of a matrix A.
Show that = Tr(B^T,A) defines an inner product on
Let U be the set of symmetric 2 x 2 matrices with real entries. U is subspace of R (2x2). Find an orthonormal basis of U with respect to the above inner product.
Let A = (aij), B = (bij) be nxn sq. matrices over R. ( means summation)
Define (A,B) = =

(k=1,2..,n)(i=1,2..,n)

You have to check the definition of inner product (as symmetric
, linear, positive definite, all very easy)

Note the dim of the vector space = 4.
And dim U = 3 (why?)
Let A=
(1 0)
(0 0)
B=
(0 0)
(0 1)
C =
(0 )
( 0)
then {A,B,C} forms an o.n. basis of U.
You should test (A,A)=(B,B)=(C,C)= 1.
(A,B) = (B,C) =(C,A) = 0
Kenny

RELATED QUESTIONS
Let A be an m x n matrix. Prove that if B can be obtained from A by an elementary row... (answered by khwang)

Let M22 have the inner product ⟨A, B⟩ = tr(A^T B). Describe the orthogonal complement (answered by ikleyn)

Let T, U: V to W be linear transformations. Prove that: a. R(T+U) is a subset of... (answered by khwang)

Let vector space M22 have the inner product defined as tr(U^T V ). Find the angle... (answered by ikleyn)

Let A ∈ Mn×n(F), and let T : Mn×n(F) → Mn×n(F) be the linear transformation X →... (answered by CPhill)

Let vector space M22 have the inner product defined as tr(U^T V ). Find ∥A∥ Where A= (answered by ikleyn)

Let X, Y and Z be matrices of sizes 2x3, 4x3 and 1x4, respectively. Then the size of... (answered by jim_thompson5910)

Need help: The lists below show five agricultural crops in Alabama, Arkansas, and... (answered by stanbon)

On the vector space M2,2 of 2*2 real matrices, show that defining =... (answered by venugopalramana)