For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the columns of A are linearly independent, then the columns of A span R. Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must span R. O B. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must span R. OC. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A do not span R. O D. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A must span R.

College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then
"statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the
implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the columns of A are linearly independent, then the columns of A span R".
Choose the correct answer below.
O A. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must
span R.
O B. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must
span R.
O C. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A do not
span R.
D. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A must
span R.
Transcribed Image Text:For this exercise assume that the matrices are all nxn. The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer. If the columns of A are linearly independent, then the columns of A span R". Choose the correct answer below. O A. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must span R. O B. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly dependent, then the columns of A must span R. O C. The statement is false. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A do not span R. D. The statement is true. By the Invertible Matrix Theorem, if the columns of A are linearly independent, then the columns of A must span R.
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