1. For each of the following statements, indicate whether they are true or false. If a statement is true, prove it. If a statement is false, give a counterexample. Let n and m be non-negative integers. (a) Let T P Pm be a linear transformation. Let [T]ε,B be the matrix of T with respect to some basis B for Pn and some basis & for Pm. If [T]ε,B has rank n + 1, then T is one-to-one. (b) Let V Mat(n, n) be the vector space of n x n-matrices. Let WC V be the subset of invertible nxn-matrices. Then W is a subspace of V. (c) The span of any two distinct nonzero vectors in R2 has dimension two. (d) Any three nonzero polynomials of distinct degrees are linearly independent in P7.
1. For each of the following statements, indicate whether they are true or false. If a statement is true, prove it. If a statement is false, give a counterexample. Let n and m be non-negative integers. (a) Let T P Pm be a linear transformation. Let [T]ε,B be the matrix of T with respect to some basis B for Pn and some basis & for Pm. If [T]ε,B has rank n + 1, then T is one-to-one. (b) Let V Mat(n, n) be the vector space of n x n-matrices. Let WC V be the subset of invertible nxn-matrices. Then W is a subspace of V. (c) The span of any two distinct nonzero vectors in R2 has dimension two. (d) Any three nonzero polynomials of distinct degrees are linearly independent in P7.
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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![1. For each of the following statements, indicate whether they are true or false. If a statement is true, prove
it. If a statement is false, give a counterexample.
Let n and m be non-negative integers.
(a) Let T P Pm be a linear transformation. Let [T]ε,B be the matrix of T with respect to some basis
B for Pn and some basis & for Pm. If [T]ε,B has rank n + 1, then T is one-to-one.
(b) Let V Mat(n, n) be the vector space of n x n-matrices. Let WC V be the subset of invertible
nxn-matrices. Then W is a subspace of V.
(c) The span of any two distinct nonzero vectors in R2 has dimension two.
(d) Any three nonzero polynomials of distinct degrees are linearly independent in P7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb15d15d9-e398-4283-aed7-d6fa30a19f66%2F253278f7-6ce5-4447-bc88-6e8524a38d13%2F2be6t6r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. For each of the following statements, indicate whether they are true or false. If a statement is true, prove
it. If a statement is false, give a counterexample.
Let n and m be non-negative integers.
(a) Let T P Pm be a linear transformation. Let [T]ε,B be the matrix of T with respect to some basis
B for Pn and some basis & for Pm. If [T]ε,B has rank n + 1, then T is one-to-one.
(b) Let V Mat(n, n) be the vector space of n x n-matrices. Let WC V be the subset of invertible
nxn-matrices. Then W is a subspace of V.
(c) The span of any two distinct nonzero vectors in R2 has dimension two.
(d) Any three nonzero polynomials of distinct degrees are linearly independent in P7.
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