Question

In: Advanced Math

Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M You need to show that (1)M...

Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M

You need to show that (1)M = QR where R := Transpose(Q)M and (2) that R is upper triangular.

To show (1) use the fact that QTranspose(Q) is the matrix for orthogonal projection onto the image of M. What happens to a column of M (which is a vector in the image of M) when you project it onto the image of M?

To show (2), think about the entries of R := Transpose(Q)M as dot products between the columns v_1,...,v_n of M and the rows u_1,...,u_n of Q^T. Entries of Transpose(Q)M vanish when these vectors are orthogonal. The vectors u_1,..., u_n are the othonomal basis for the image of M obtained from v_1,...,v_n via the Gram-Schmidt process. Why is it the case that u_i.v_j =0 if i>j?

Solutions

Expert Solution


Related Solutions

Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then...
Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then P = Q.
Let A = {r belongs to Q : e < r < pi} show that A...
Let A = {r belongs to Q : e < r < pi} show that A is closed and bounded but not compact
Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements...
Consider an orthogonal matrix Q ∈ R ^m×m and a diagonal matrix with non-zero diagonal elements D ∈ R^ m×m and answer the True— False questions below. (k) Any vector x ∈ R^m can be written as a linear combination of the columns of D (l) The columns of Q span R^m (m) The columns of D span R^m (n) The columns of Q are a basis for R^m (o) The columns of D are a basis for R^m
6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the...
6.4.13. If R is the ring of Gaussian integers, show that Q(R) is isomorphic to the subfield of C consisting of complex numbers with rational real and imaginary parts.
You will need to use R to compute the final probabilities, but show all of the...
You will need to use R to compute the final probabilities, but show all of the work up to that point and write down the R code that you ran to compute your final answer. Problems should be done in the order listed, and should be clear and complete for full marks. 1. The tensile strength of plastic used to make grocery bags is an important quality characteristic. It is known that the strength is normally distributed with mean, μ...
Write a truth table for the proposition: ¬(q ∧ r) → (¬p ∨ ¬r). Consider a “1” to be true and a “0” to be false.
Write a truth table for the proposition: ¬(q ∧ r) → (¬p ∨ ¬r). Consider a “1” to be true and a “0” to be false.
Instructions This assignment is to be typed up in the supplied R-Script. You need to show...
Instructions This assignment is to be typed up in the supplied R-Script. You need to show all of your work in R in the given script. Be sure and use # in front of any text that you type in. You are allowed to work with your peer group, but please site your sources! If you get help from anyone, you need to mention that in your write up. This assignment is worth 60 points (10 per problem) and you...
Instructions This assignment is to be typed up in the supplied R-Script. You need to show...
Instructions This assignment is to be typed up in the supplied R-Script. You need to show all of your work in R in the given script. Be sure and use # in front of any text that you type in. You are allowed to work with your peer group, but please site your sources! If you get help from anyone, you need to mention that in your write up. This assignment is worth 60 points (10 per problem) and you...
Instructions This assignment is to be typed up in the supplied R-Script. You need to show...
Instructions This assignment is to be typed up in the supplied R-Script. You need to show all of your work in R in the given script. 3. Infant mortality. The infant mortality rate is defined as the number of infant deaths per 1,000 live births. This rate is often used as an indicator of the level of health in a country. The relative frequency histogram below shows the distribution of estimated infant death rates for 224 countries for which such...
4. Show that the set A = {fm,b : R → R | m does not...
4. Show that the set A = {fm,b : R → R | m does not equal 0 and fm,b(x) = mx + b, m, b ∈ R} forms a group under composition of functions. (The set A is called the set of affine functions from R to R.)
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT