{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Practice Problems

Practice Problems - with reasoning based on theorems...

This preview shows pages 1–2. Sign up to view the full content.

Math 136 Practice Problems # 2 1. Which of the following sets are linearly independent? If a set is linearly dependent, give a non-trivial linear combination of the vectors which equals the zero vector. a) S = 2 1 3 , 1 0 - 1 . b) T = 1 0 1 , - 2 0 - 2 , 1 1 1 . c) U = 0 0 0 , 1 1 - 1 , - 1 - 1 1 . 2. Determine, with proof, which of the following are subspaces of R 3 and which are not. a) S 1 = x 1 x 2 x 3 R 3 | x 1 = - x 2 b) S 2 = x 1 x 2 x 3 R 3 | x 1 - 2 x 2 + 3 x 3 = - 1 c) S 3 = x 1 x 2 x 3 R 3 | x 1 ,x 2 ,x 3 Z d) S 4 = x 1 x 2 x 3 R 3 | x 1 + x 2 = 0 ,x 3 = - x 2 e) S 5 = x 1 x 2 x 3 R 3 | x 1 + x 1 x 3 + x 2 = 0 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 3. Suppose that the set of vectors { ~ v 1 , ~ v 2 , ~ v 3 ,..., ~ v k } is linearly independent, and that the set of vectors { ~ u 1 , ~ u 2 , ~ u 3 ,..., ~ u } is linearly dependent, prove that the set of vectors [ { 1 ~ v 1 , 2 ~ v 2 ,...,k~ v k , - 1 ~ u 1 , - 2 ~ u 2 ,..., - ‘~ u } is linearly dependent. 4. Determine whether each of the following statements is true or false. Justify your choices
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: with reasoning based on theorems, deﬁnitions, or counter-examples. a) If ~v 1 is a vector in R n , then the set { ~v 1 } is linearly independent. b) If { ~v 1 ,...,~v k } is a linearly dependent set in R n , then { ~v 1 ,...,~v k-1 } is linearly dependent. c) If { ~v 1 ,~v 2 } is a linearly independent set in R n , then { ~v 1 ,~v 1-~v 2 } is linearly independent. * 5. Let ~x = ± a b ² , ~ y = ± c d ² , and ~ z = ± e f ² be any three vectors in R 2 . a) Prove that { ~x,~ y } is linearly independent if and only if ad-bc 6 = 0. b) Prove that { ~x,~ y,~ z } is linearly dependent. * Indicates a challenging problem....
View Full Document

{[ snackBarMessage ]}

Page1 / 2

Practice Problems - with reasoning based on theorems...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online