Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that V1, V2, V3 is a basis for Uj and V4, V5, V6 is a basis for U2. Suppose that Uin U2 = 0. Prove that В - B = v1, V2, V3, V4, V5, V6 is a basis for U1 + U2.
Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that V1, V2, V3 is a basis for Uj and V4, V5, V6 is a basis for U2. Suppose that Uin U2 = 0. Prove that В - B = v1, V2, V3, V4, V5, V6 is a basis for U1 + U2.
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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![Let Uj and U2 be vector subspaces of a
vector space V over the real numbers.
Suppose that V1, V2, V3 is a basis for U1 and
V4, V5, V6 is a basis for U2. Suppose that
U1NU2 = 0. Prove that
B = v1, V2, V3, V4, V5, V6 is a basis for
U1+ U2.
Suppose now that V = U1 0 U2. Let A be
the set of all linear transformations
T :V → V such that T(U1) C Uj and
T(U2) C U2.
Prove that A is a subspace of the set of all
linear transformations
L(V) = {T : V → V,T is a linear transform.
Give an example of a matrix A =B [T]B_that
may occur as the matrix of a linear
transformation T E A. Pay particular
attention to the block matrix structure of your
example matrix A. You do not need to
provide the linear transformation T
corresponding to A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23cdc0d7-7d33-42d3-8fb9-b8ab1ea99c34%2Fa66c6a8d-db21-41b7-b1f9-2a882eeae076%2Fkjmwz5k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let Uj and U2 be vector subspaces of a
vector space V over the real numbers.
Suppose that V1, V2, V3 is a basis for U1 and
V4, V5, V6 is a basis for U2. Suppose that
U1NU2 = 0. Prove that
B = v1, V2, V3, V4, V5, V6 is a basis for
U1+ U2.
Suppose now that V = U1 0 U2. Let A be
the set of all linear transformations
T :V → V such that T(U1) C Uj and
T(U2) C U2.
Prove that A is a subspace of the set of all
linear transformations
L(V) = {T : V → V,T is a linear transform.
Give an example of a matrix A =B [T]B_that
may occur as the matrix of a linear
transformation T E A. Pay particular
attention to the block matrix structure of your
example matrix A. You do not need to
provide the linear transformation T
corresponding to A.
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