(a) T F If a set of m vectors in R" is linearly independent, then m≤n. (b) T F A5x8 matrix cannot have a 2-dimensional null space. (c) T F Ifa 5 x 5 matrix has only 3 distinct eigenvalues, then A cannot be diagonalizable.
(a) T F If a set of m vectors in R" is linearly independent, then m≤n. (b) T F A5x8 matrix cannot have a 2-dimensional null space. (c) T F Ifa 5 x 5 matrix has only 3 distinct eigenvalues, then A cannot be diagonalizable.
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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Answer true or false. Please answer the first 6
![6. (a) T F If a set of m vectors in R" is linearly independent, then m≤n.
(b) T F A5 x 8 matrix cannot have a 2-dimensional null space.
(c) T F If a 5 x 5 matrix has only 3 distinct eigenvalues, then A cannot be diagonalizable.
(d) T F
A linear transformation from R³ to R³ cannot be one-to-one.
(e) T F If r¹ Ar> 0 for some r, then the quadratic form ¹ Ar is positive definite.
(f) T F If a linear system Ax = b is consistent, then rank [Ab] = rank A.
(g) T F If two square matrices are similar, then they have the same eigenvalues.
(h) T F If W is a subspace of R³ and v is a vector in R³, then v is in W or v is in W+.
(i) T F If U is an orthogonal matrix, then the rows of U form an orthonormal basis of R".
(j) T F If A is a 3 x 3 symmetric matrix and if u and v are vectors in R³ such that
Au = 3u and Av = 5v, then u. v = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66b4274c-507e-4eb6-af25-90299dfa30b6%2F799e2430-51bb-4261-b6f6-444cc389ea3b%2Felwkv46_processed.png&w=3840&q=75)
Transcribed Image Text:6. (a) T F If a set of m vectors in R" is linearly independent, then m≤n.
(b) T F A5 x 8 matrix cannot have a 2-dimensional null space.
(c) T F If a 5 x 5 matrix has only 3 distinct eigenvalues, then A cannot be diagonalizable.
(d) T F
A linear transformation from R³ to R³ cannot be one-to-one.
(e) T F If r¹ Ar> 0 for some r, then the quadratic form ¹ Ar is positive definite.
(f) T F If a linear system Ax = b is consistent, then rank [Ab] = rank A.
(g) T F If two square matrices are similar, then they have the same eigenvalues.
(h) T F If W is a subspace of R³ and v is a vector in R³, then v is in W or v is in W+.
(i) T F If U is an orthogonal matrix, then the rows of U form an orthonormal basis of R".
(j) T F If A is a 3 x 3 symmetric matrix and if u and v are vectors in R³ such that
Au = 3u and Av = 5v, then u. v = 0.
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