Let A denote the 2 x 2 matrix We define a function (, ) : R² ×x R² → R by taking (x, y) = y" Ax, for each pair of vectors x, y E R². (Technically, the result of computing y Ax is a 1 x 1 matrix, not a real number. However, we identify the 1 x 1 matrix with its single real entry.) (a) Does (, ·) define an inner product on R? If it does, prove it. If it does not, state which inner product conditions this function satisfies, and prove those. Also, provide counterexamples for each inner product condition that this function does not satisfy. (b) By replacing A with another matrix B different from A in only a single entry, define an inner product [', ] different from the function in part (a), and justify why this modified function is an inner product. (c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ] that you chose in part (b). Justify that your chosen basis is orthonormal.
Let A denote the 2 x 2 matrix We define a function (, ) : R² ×x R² → R by taking (x, y) = y" Ax, for each pair of vectors x, y E R². (Technically, the result of computing y Ax is a 1 x 1 matrix, not a real number. However, we identify the 1 x 1 matrix with its single real entry.) (a) Does (, ·) define an inner product on R? If it does, prove it. If it does not, state which inner product conditions this function satisfies, and prove those. Also, provide counterexamples for each inner product condition that this function does not satisfy. (b) By replacing A with another matrix B different from A in only a single entry, define an inner product [', ] different from the function in part (a), and justify why this modified function is an inner product. (c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ] that you chose in part (b). Justify that your chosen basis is orthonormal.
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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Plz provide full solutions to all these questions. Thx!
![1.
2. Let A denote the 2 x 2 matrix
(: )
2
We define a function (, ·) : R² × R² → R by taking
1
(х, у) — у" Ах,
for each pair of vectors x, y E R?. (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a
real number. However, we identify the 1 x 1 matrix with its single real entry.)
(a) Does (, ) define an inner product on R?? If it does, prove it. If it does not, state which inner
product conditions this function satisfies, and prove those. Also, provide counterexamples for
each inner product condition that this function does not satisfy.
(b) By replacing A with another matrix B different from A in only a single entry, define an inner
product [, ] different from the function in part (a), and justify why this modified function is an
inner product.
(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ] that you
chose in part (b). Justify that your chosen basis is orthonormal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18ec0807-f91e-438f-98bb-9092cc352bac%2F270ab567-7dfe-4737-9585-da497d67d123%2Fsv84k6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
2. Let A denote the 2 x 2 matrix
(: )
2
We define a function (, ·) : R² × R² → R by taking
1
(х, у) — у" Ах,
for each pair of vectors x, y E R?. (Technically, the result of computing y" Ax is a 1 x 1 matrix, not a
real number. However, we identify the 1 x 1 matrix with its single real entry.)
(a) Does (, ) define an inner product on R?? If it does, prove it. If it does not, state which inner
product conditions this function satisfies, and prove those. Also, provide counterexamples for
each inner product condition that this function does not satisfy.
(b) By replacing A with another matrix B different from A in only a single entry, define an inner
product [, ] different from the function in part (a), and justify why this modified function is an
inner product.
(c) Give an orthonormal basis for R? with respect to this newly-defined inner product [, ] that you
chose in part (b). Justify that your chosen basis is orthonormal.
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