Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial isometry if there exists a subspace W of V such that ||U(x)|| = ||x|| for all x EW and U(x) = 0 for all x € W¹. Observe that W need not be U-invariant. Suppose that U is such an operator and {V₁, V2, ..., Uk} is an orthonormal basis for W. Prove the following results. (b) {U(v₁), U(v₂),..., U(Uk)} is an orthonormal basis for R(U).
Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial isometry if there exists a subspace W of V such that ||U(x)|| = ||x|| for all x EW and U(x) = 0 for all x € W¹. Observe that W need not be U-invariant. Suppose that U is such an operator and {V₁, V2, ..., Uk} is an orthonormal basis for W. Prove the following results. (b) {U(v₁), U(v₂),..., U(Uk)} is an orthonormal basis for R(U).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let V be a finite-dimensional inner product space. A linear operator U
on V is called a partial isometry if there exists a subspace W of V
such that ||U(x)|| = ||x|| for all x Є W and U(x) = 0 for all x € W¹.
Observe that W need not be U-invariant. Suppose that U is such an
operator and {V₁, V2, ..., Uk} is an orthonormal basis for W. Prove the
following results.
(b) {U(v₁), U(v₂),..., U(Uk)} is an orthonormal basis for R(U).
![(c) There exists an orthonormal basis for V such that the first
k columns of [U], form an orthonormal set and the remaining
columns are zero.
-
(d) Let {w₁, W2, . ‚w;} be an orthonormal basis for R(U)± and ß =
{U(v₁), U(v₂),..., U(Vk), W₁, ..., W;}. Then 3 is an orthonormal
basis for V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc84dbade-4701-4cff-b1a4-c3a634deeabf%2F8316651e-5c11-4ac2-b938-e1c537a432b1%2F3gcj33_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(c) There exists an orthonormal basis for V such that the first
k columns of [U], form an orthonormal set and the remaining
columns are zero.
-
(d) Let {w₁, W2, . ‚w;} be an orthonormal basis for R(U)± and ß =
{U(v₁), U(v₂),..., U(Vk), W₁, ..., W;}. Then 3 is an orthonormal
basis for V.
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