4. Let T : R" → R" be a linear transformation that preserves lengths;that is||T(r)|| = ||æ|| for all x e R".(a) Show that T preserves orthogonality.(b) Show that the standard matrix of [A] of T is an orthogonal matrix.

Answered: Image /qna-images/answer/6e155d7d-8183-40ea-a731-a6e7b922f0c4.jpg

Answered: 4. Let T : R" that is||T(x)|| = ||æ||…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Let T : R³ → R³ be linear transformation. The sequence O ) (v1, v2, v3) = ( |) is a basis of R3. If T(v1) = |2),T(v2) = | 3,T(v3) = determine the standard matrix of T.
let The Linear Transformation given by T:P3toP2 T(ax³+bx²+cx+d)) = 3 ax²+2bx c find a basis for ker(T) and a basis for range(T). Also verify the dim ker(T) + dim {range(T)} = dim v
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
Let T: R¹ → R" be a linear transformation, and let {₁,...,} be a set vectors in Rr. Prove that if {T(v₁),...,T(up)} is linearly independent, then {₁,...,Up} is linearly independent.
A function T:VW between vector spaces which satisfies T(a+b) = T(a) + T(b) for all a, b eV must be a linear transformation. Select one O True False
'
How large should n be to guarantee that the Trapezoidal Rule approximation to x - 12x- 48x² + 2x + 5) dæ is accurate to within 0.1. n = How large should n be to guarantee that the Simpsons Rule approximation to 2 1(-a - 12a - 48z? + 2x + 5) dx is accurate to within 0.1. n = Hint: Remember your answers should be a whole numbers, and Simpson's Rule requires even values for n Question Help: Video 1 D Video 2 Submit Question P Type here to search hp
4. Let V be a vector space. Prove that a) The zero transformation T(v) = b) The identity transformation T(v) = v for all v EV is a linear transformation. = 0 for all v EV is a linear transformation. CS Scanned with CamScanner
Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.
Consider linear transformation F:R4 R³ given by F(w, x, y, z)=(w-3x+2y-z, x-y+z, w+y-z) a) b) c) d) Find the standard matrix A for the transformation F. Find the basis kernel of F. Find the nullity (F) and rank (F). Is F one-to-one? Give a reason for your answer.
1. Let W be the line y = x in R². (a) Find a unit vector u on W. (b) Consider the linear transformation T: R² R² that projects each vector orthogonally onto W. Calculate the matrix for T which is uu¹. (c) Use that matrix to calculate T (d) Draw the vector [] [4] T[A] the line W, and the projection T
defined by Let Let ƒ : R² → R² be the linear transformation [f] B = B C f(x) = = = 3 2 -2 - -3 x. be two different bases for R². Find the matrix [f] for f relative to the basis B in the domain and C in the codomain. {(-1,2), (3,-7)}, {{-1, -2), (3, 7)},

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

4. Let T : R" that is||T(x)|| = ||æ|| for all re R". R" be a linear transformation that preserves lengths; (a) Show that T preserves orthogonality. (b) Show that the standard matrix of [A] of T is an orthogonal matrix.