practiceprob2_soln

practiceprob2_soln - Math 235 Assignment 2 Practice...

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

View Full Document Right Arrow Icon
Math 235 Assignment 2 Practice Problems Solutions 1. Consider the projection proj ( - 2 , 3) : R 2 R 2 onto the line ~x = t ± - 2 3 ² , t R . Determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . Solution: Pick ~v 1 = ± - 2 3 ² . We then pick ~v 2 = ± 3 2 ² so that it is orthogonal to ~v 1 . By geometrical arguments, a basis adapted to proj ~v 1 is B = { ~v 1 ,~v 2 } . To determine the matrix of proj (3 , 2) with respect to B , calculate the B coordinates of the images of the basis vectors: proj ~v 1 ( ~v 1 ) = ~v 1 = 1 ~v 1 + 0 ~v 2 proj ~v 1 ( ~v 2 ) = ~ 0 = 0 ~v 1 + 0 ~v 2 Hence, we get [proj ~v 1 ] B = ± 1 0 0 0 ² . 2. Let U , V , W be finite dimensional vector spaces over R and let L : V U and M : U W be linear mappings. a) Prove that rank( M L ) rank( M ). Solution: Since the rank of a linear mapping is equal to the dimension of the range, we consider the range of both mappings. Observe that every vector of the form ( M L
Background image of page 1

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

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

This note was uploaded on 11/28/2011 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

Page1 / 2

practiceprob2_soln - Math 235 Assignment 2 Practice...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online