Math416_HW10_sol - Math 416 Abstract Linear Algebra Fall...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Homework 10 solutions Section 5.5 5.2. (1 pt check) Since A has rank 2, let us find the orthogonal projections onto the 1-dimensional subspaces Null A and Null A T . A = 1 1 1 1 3 2 2 4 3 ∼ 1 1 1 0 2 1 0 2 1 ∼ 1 1 1 0 2 1 0 0 0 ∼ 2 2 2 0 2 1 0 0 0 ∼ 2 0 1 0 2 1 0 0 0 Null A has basis v = 1 1- 2 . The projection matrix onto Null A is therefore Proj Null A = v ( v T v )- 1 v T = 1 1- 2 6- 1 1 1- 2 = 1 6 1 1- 2 1 1- 2- 2- 2 4 The projection onto Row A = (Null A ) ⊥ is Proj Row A = I- Proj Null A = 1 6 6 0 0 0 6 0 0 0 6 - 1 1- 2 1 1- 2- 2- 2 4 = 1 6 5- 1 2- 1 5 2 2 2 2 . Likewise, let us find a basis of Null A T : A T = 1 1 2 1 3 4 1 2 3 ∼ 1 1 2 0 2 2 0 1 1 ∼ 1 1 2 0 1 1 0 0 0 ∼ 1 0 1 0 1 1 0 0 0 . 1 Null A T has basis w = 1 1- 1 . The projection matrix onto Null A T is therefore Proj Null A T = w ( w T w )- 1 w T = 1 1- 1 3- 1 1 1- 1 = 1 3 1 1- 1 1 1- 1- 1- 1 1 . The projection onto Col A = (Null A T ) ⊥ is Proj Col A = I- Proj Null A T = 1 3 3 0 0 0 3 0 0 0 3 - 1 1- 1 1 1- 1- 1- 1 1 = 1 3 2- 1 1- 1 2 1 1 1 2 . 5.4. (2 pts) a. The rank-nullity theorem applied to A and A * A yields ( rank A + dim ker A = n rank( A * A ) + dim ker( A * A ) = n ⇒ rank A + dim ker A = rank( A * A ) + dim ker( A * A ) ⇒ rank A = rank( A * A ) since ker A = ker( A * A ) . b. If A has trivial kernel, then so does A * A : ker( A * A ) = ker A = { } . Since A * A is square (i.e. its domain and codomain have the same dimension) and injective, it is invertible. Using the inverse of A * A , we obtain ( A * A )- 1 A * A = I so that ( A * A )- 1 A * is a left inverse of A ....
View Full Document

{[ snackBarMessage ]}

Page1 / 9

Math416_HW10_sol - Math 416 Abstract Linear Algebra Fall...

This preview shows document pages 1 - 3. Sign up to view the full document.

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