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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.

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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 1EQ
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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.
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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