Let TR4 R³ be given by T(v) = Av, where 1 1 1 2 (334 1 1 1 3 0004 A = Find a basis {u₁, u₂} of ker T, and extend it to a basis {V₁, V2, U₁1, U₂} of R4. Compute w₁ = T(v₁) and w₂ = T(v₂). Find a basis {W₁, W2,Z₁} of R³. Find the matrix B of T with respect to the basis {V₁, V2, U₁, U₂} of R4 and the basis {w₁, W2, Z1} of R3. Verify that B is in canonical form.
Let TR4 R³ be given by T(v) = Av, where 1 1 1 2 (334 1 1 1 3 0004 A = Find a basis {u₁, u₂} of ker T, and extend it to a basis {V₁, V2, U₁1, U₂} of R4. Compute w₁ = T(v₁) and w₂ = T(v₂). Find a basis {W₁, W2,Z₁} of R³. Find the matrix B of T with respect to the basis {V₁, V2, U₁, U₂} of R4 and the basis {w₁, W2, Z1} of R3. Verify that B is in canonical form.
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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Question
can you please provide explanantions
![Let TR4R³ be given by T(v) = Av, where
1 1 1 2
1
1 3
1
0004
A =
Find a basis {u₁, u₂} of ker T, and extend it to a basis {V₁, V2, U1₁, U₂) of R4.
Compute w₁ = T(v₁) and W₂ = T(v₂). Find a basis {W₁, W2, Z₁} of R³.
Find the matrix B of T with respect to the basis {V₁, V2, U₁, U₂} of R¹ and the
basis {w₁, W2,Z₁} of R³. Verify that B is in canonical form.
By considering transition matrices, write down invertible matrices P and such
that BQ-¹AP. Check that QB = AP.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4061eb31-6ba2-4539-9b51-dd7b05481e7e%2F3aabf92a-5fa6-4f7d-b363-88e56378b2a1%2Fm1btc4o_processed.png&w=3840&q=75)
Transcribed Image Text:Let TR4R³ be given by T(v) = Av, where
1 1 1 2
1
1 3
1
0004
A =
Find a basis {u₁, u₂} of ker T, and extend it to a basis {V₁, V2, U1₁, U₂) of R4.
Compute w₁ = T(v₁) and W₂ = T(v₂). Find a basis {W₁, W2, Z₁} of R³.
Find the matrix B of T with respect to the basis {V₁, V2, U₁, U₂} of R¹ and the
basis {w₁, W2,Z₁} of R³. Verify that B is in canonical form.
By considering transition matrices, write down invertible matrices P and such
that BQ-¹AP. Check that QB = AP.
=
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