{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sol34 - Partial Solution Set Leon 3.4 3.4.3 Given the...

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

View Full Document Right Arrow Icon
Partial Solution Set, Leon § 3.4 3.4.3 Given the vectors x 1 = (2 , 1) T , x 2 = (4 , 3) T , and x 3 = (7 , - 3) T , (a) Show that x 1 and x 2 form a basis for R 2 . (b) Why must x 1 , x 2 , and x 3 be linearly dependent? (c) What is the dimension of Span( x 1 , x 2 , x 3 )? Solution : (a) This follows because they are (by inspection) linearly independent in R 2 . Since the dimension of R 2 is 2, x 1 and x 2 form a basis for R 2 . (b) Since R 2 is 2-dimensional, any collection of more than two vectors from R 2 must be linearly dependent. (c) Since x 1 and x 2 form a basis for R 2 , the dimension of Span( x 1 , x 2 , x 3 ) is 2, i.e., Span( x 1 , x 2 , x 3 ) = R 2 . 3.4.4 Given the vectors, x 1 = (3 , - 2 , 4) T , x 2 = ( - 3 , 2 , - 4) T , and x 3 = ( - 6 , 4 , - 8) T , what is the dimension of Span( x 1 , x 2 , x 3 )? Solution : Both x 2 and x 3 are scalar multiples of x 1 , so the dimension of Span( x 1 , x 2 , x 3 ) is 1. 3.4.5 Given the vectors, x 1 = (2 , 1 , 3) T , x 2 = (3 , - 1 , 4) T , and x 3 = (2 , 6 , 4) T , (a) Show that x 1 , x 2 , and x 3 are linearly dependent. (b) Show that x 1 and x 2 are linearly independent. (c) What is the dimension of Span( x 1 , x 2 , x 3 )? (d) Give a geometric description of Span( x 1 , x 2 , x 3 ). Solution : (a) Letting A = x 1 x 2 x 3 , we consider the solutions to A x = 0 . (Yes, this is one of those situations in which the matrix turns out to be square, so the determinant is a possibility. But it is not recommended. Use Gaussian elimination instead.)
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}