{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

practiceprob2_soln

# practiceprob2_soln - Math 235 Assignment 2 Practice...

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

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 , 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 . We then pick ~v 2 = 3 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. 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 )( ~x ) = M ( L ( ~x )) can be written as M ( ~ y ) for some ~ y U . Thus, the range of M L is a subspace of the range of M . Hence, the dimension of the range of

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}