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1. Consider the vector space R2 with the norm || - ||p. For p = 1,2,...
c and d only 2. Consider the vector space R3 with the standard inner product and the standard norm |x| x, x) Use the fo...
c and d only
2. Consider the vector space R3 with the standard inner product and the standard norm |x| x, x) Use the formula for projection given in Chapter 5, Section 4.2 of LADW to find the matrix of orthogonal projection P onto the column space of the matrix -) 1 1 A = 2 4 (a) What is the projection matrix P? (b) What is the size of P? (c) Since the dimension of the column space of...
is some norm on a vector spad then there is some inner product such that 1||...
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
5. For parts (a)-(d) below, consider the set of vectors B = {(1,2), (2, -1)}. (a)...
5. For parts (a)-(d) below, consider the set of vectors B = {(1,2), (2, -1)}. (a) (2 points) Demonstrate that B is an orthogonal set in the Euclidean inner product space R2. (b) (3 points) Use the set B to create an orthonormal basis in the Euclidean inner product space R2 (e) (7 points) Find the transition matrix from the standard basis S = {(1,0),(0,1)} for R2 to the basis B. Show all steps in your calculation. (d) (7 points)...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,)...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product def...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) =...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B.
4. Consider the vector space...
3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2 and the linear map T :R2[x]−→R2[x], T(p...
3. Consider the vector space V = R2[x] with its standard ordered basisE = 1,x,x2 and the linear map T :R2[x]−→R2[x], T(p)=p(x−1)−p(0)x2 (a) (1 point) What is [T]E? (b) (1 point) Is T invertible? (c) (6 points) Compute the eigenvalues of T and their algebraic multiplicity. (d) (2 points) Is T diagonalisable? If so, find a matrix Q such that Q−1[T]EQ is diagonal. If not, findQ, so that the above matrix is upper triangular.
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 → R2 be the reflection operator with respect...
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 → R2 be the reflection operator with respect to the x-axis, defined by 21 21 Compute the adjoint operator Lt. Is L self-adjoint?
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric
Let L: R2 →...
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of ...
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point (not the origin) on M. Draw the tangential plane to N at p. Moreover, draw all null vectors with origin at p.
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incompl...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
c and d only
2. Consider the vector space R3 with the standard inner product and the standard norm |x| x, x) Use the formula for projection given in Chapter 5, Section 4.2 of LADW to find the matrix of orthogonal projection P onto the column space of the matrix -) 1 1 A = 2 4 (a) What is the projection matrix P? (b) What is the size of P? (c) Since the dimension of the column space of...
is some norm on a vector spad then there is some inner product such that 1|| ||is some norm on a vector space V, then there is some inner product (:, - ) such that ||0||2 = (v, v) for all v E V. Select one: O True False
5. For parts (a)-(d) below, consider the set of vectors B = {(1,2), (2, -1)}. (a) (2 points) Demonstrate that B is an orthogonal set in the Euclidean inner product space R2. (b) (3 points) Use the set B to create an orthonormal basis in the Euclidean inner product space R2 (e) (7 points) Find the transition matrix from the standard basis S = {(1,0),(0,1)} for R2 to the basis B. Show all steps in your calculation. (d) (7 points)...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
let P3 denote the vector space of polynomials of degree 3 or
less, with an inner product defined by
14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements
14. Let Ps denote the vector space of polynomials of degree 3 or less,...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B.
4. Consider the vector space...
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 → R2 be the reflection operator with respect to the x-axis, defined by 21 21 Compute the adjoint operator Lt. Is L self-adjoint?
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric
Let L: R2 →...
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point (not the origin) on M. Draw the tangential plane to N at p. Moreover, draw all null vectors with origin at p.
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
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