Question

Let V be the vector space of 2 × 2 matrices over R, let <A, B>=...

Let V be the vector space of 2 × 2 matrices over R, let <A, B>= tr(ABT ) be an inner product on V , and let U ⊆ V be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal projection of the matrix A = (1 2

3 4)

on U, and compute the minimal distance between A and an element of U.

Hint: Use the basis 1 0 0 0  

0 0 0 1  

0 1 1 0 of U.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
A2. Let v be a fixed vector in an inner product space V. Let W be...
A2. Let v be a fixed vector in an inner product space V. Let W be the subset of V consisting of all vectors in V that are orthogonal to v. In set language, W = { w LaTeX: \in ∈V: <w, v> = 0}. Show that W is a subspace of V. Then, if V = R3, v = (1, 1, 1), and the inner product is the usual dot product, find a basis for W.
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Let W be a subspace of a f.d. inner product space V and let PW be...
Let W be a subspace of a f.d. inner product space V and let PW be the orthogonal projection of V onto W. Show that the characteristic polynomial of PW is (t-1)^dimW t^(dimv-dimw)
1. Find the orthogonal projection of the matrix [[3,2][4,5]] onto the space of diagonal 2x2 matrices...
1. Find the orthogonal projection of the matrix [[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form lambda?I.   [[4.5,0][0,4.5]]  [[5.5,0][0,5.5]]  [[4,0][0,4]]  [[3.5,0][0,3.5]]  [[5,0][0,5]]  [[1.5,0][0,1.5]] 2. Find the orthogonal projection of the matrix [[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace 0.   [[-1,3][3,1]]  [[1.5,1][1,-1.5]]  [[0,4][4,0]]  [[3,3.5][3.5,-3]]  [[0,1.5][1.5,0]]  [[-2,1.5][1.5,2]]  [[0.5,4.5][4.5,-0.5]]  [[-1,6][6,1]]  [[0,3.5][3.5,0]]  [[-1.5,3.5][3.5,1.5]] 3. Find the orthogonal projection of the matrix [[1,5][1,2]] onto the space of anti-symmetric 2x2 matrices.   [[0,-1] [1,0]]  [[0,2] [-2,0]]  [[0,-1.5] [1.5,0]]  [[0,2.5] [-2.5,0]]  [[0,0] [0,0]]  [[0,-0.5] [0.5,0]]  [[0,1] [-1,0]] [[0,1.5] [-1.5,0]]  [[0,-2.5] [2.5,0]]  [[0,0.5] [-0.5,0]] 4. Let p be the orthogonal projection of u=[40,-9,91]T onto the...
9. Let S and T be two subspaces of some vector space V. (b) Define S...
9. Let S and T be two subspaces of some vector space V. (b) Define S + T to be the subset of V whose elements have the form (an element of S) + (an element of T). Prove that S + T is a subspace of V. (c) Suppose {v1, . . . , vi} is a basis for the intersection S ∩ T. Extend this with {s1, . . . , sj} to a basis for S, and...
1. V is a subspace of inner-product space R3, generated by vector u =[2 2 1]T...
1. V is a subspace of inner-product space R3, generated by vector u =[2 2 1]T and v =[ 3 2 2]T. (a) Find its orthogonal complement space V┴ ; (b) Find the dimension of space W = V+ V┴; (c) Find the angle θ between u and v and also the angle β between u and normalized x with respect to its 2-norm. (d) Considering v’ = av, a is a scaler, show the angle θ’ between u and...
Let V be a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose...
Let V be a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose that T is a linear transformation from V to itself and T(u) = u + v, T(v) = u, T(w) = v. 1. Find the matrix of T relative to the ordered basis B. 2. A typical element of V looks like au + bv + cw, where a, b and c are scalars. Find T(au + bv + cw). Now that you know...
We can identify the set V of all 3×3-matrices (real coefficients) with vector space R9. Show...
We can identify the set V of all 3×3-matrices (real coefficients) with vector space R9. Show that the set of all 3 × 3 symmetric matrices is a vector subspace of V .
Let Mn be the vector space of all n × n matrices with real entries. Let...
Let Mn be the vector space of all n × n matrices with real entries. Let W = {A ∈ M3 : trace(A) = 0}, U = {B ∈ Mn : B = B t } Verify U, W are subspaces . Find a basis for W and U and compute dim(W) and dim(U).
a. Find an orthonormal basis for R 3 containing the vector (2, 2, 1). b. Let...
a. Find an orthonormal basis for R 3 containing the vector (2, 2, 1). b. Let V be a 3-dimensional inner product space and S = {v1, v2, v3} be an orthonormal set in V. Explain whether the set S can be a basis for V .
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT