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4. State with a brief explanation whether the following statements are true or false. (a) The vectors (1,0) and (0, 1) span R². (b) The vectors (2,−1), (0, 1), and (1,0) span R². (c) The vectors (1,0) and (0, 1) are linearly independent. (d) The vectors (2, -1), (0, 1) and (1,0) are linearly independent. (e) A set of any two vectors in R² is a basis for R². (f) A set of any three vectors in R² must be linearly dependent. (g) R2 is a subspace of R³.

4. State with a brief explanation whether the following statements are true or false. (a) The vectors (1,0) and (0, 1) span R². (b) The vectors (2,−1), (0, 1), and (1,0) span R². (c) The vectors (1,0) and (0, 1) are linearly independent. (d) The vectors (2, -1), (0, 1) and (1,0) are linearly independent. (e) A set of any two vectors in R² is a basis for R². (f) A set of any three vectors in R² must be linearly dependent. (g) R2 is a subspace of R³.

Advanced Engineering Mathematics
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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4. State with a brief explanation whether the following statements are true or false. (a) The vectors (1,0) and (0, 1) span R². (b) The vectors (2, -1), (0, 1), and (1,0) span R². (c) The vectors (1,0) and (0, 1) are linearly independent. (d) The vectors (2, -1), (0, 1) and (1,0) are linearly independent. (e) A set of any two vectors in R2 is a basis for R². (f) A set of any three vectors in R2 must be linearly dependent. (g) R2 is a subspace of R³.
Transcribed Image Text:4. State with a brief explanation whether the following statements are true or false. (a) The vectors (1,0) and (0, 1) span R². (b) The vectors (2, -1), (0, 1), and (1,0) span R². (c) The vectors (1,0) and (0, 1) are linearly independent. (d) The vectors (2, -1), (0, 1) and (1,0) are linearly independent. (e) A set of any two vectors in R2 is a basis for R². (f) A set of any three vectors in R2 must be linearly dependent. (g) R2 is a subspace of R³.
Expert Solution
Step 1

a) true

The vectors (1 ,0) and (0,1) is linearly independent.

We know that linearly independent set of n vectors in n dimensional vector space is a basis that span vector space 

(1,0) ,(0,1) is basis of R2 also span R2

 

 

b)true 

(2 ,-1) ,(0,1) and (1,0) span R2 

(0,1) and (1,0) togather span R2

If we add extra vectors that also span R2

 

 

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