{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

TUE7Quiz7

# TUE7Quiz7 - Introduction to Linear Algebra Tue 7 pm...

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: Introduction to Linear Algebra Tue. 7 pm Department I ID 1 Name I 1. Indicate whether the statement is true (T) or false (F). (a) If rank(A) = rank(A T), then A is a square. ( F l 100]) 010 (b) Let A be a m><n matrix, then rank(A TA) = rank(AAT). ( T ) (By Thm 7.5.8(d) & 7.5.9(d), rank(A) = rank(A TA) = rank(AA T) ) (counterexample A=[ 2. Find the column—row factorization and column—row expansion of the following matrix. 1 2 8 A: 717175 2 5 19 102 sol) r.r.e.f of A : BO: 013 000 1 2 8 1 2 102 80, A has the column—row factorization of A: —1—1—5 =CR= —1—1 [013] 2 5 19 2 5 1 2 8 1 2 And column-row expansion of A: —1—1—5 = —1 [1021+ —1 [013] 2 5 19 2 5 1 0 2 0 2 6 = —10—2 + 0—1—3 2 O 4 O 5 15 3. If A is a square matrix for which A and A2 have the same rank, then null(A) ﬂ col(A) = {0}. pf) If rank(A2) = rank(A) then dim(null(A2)) = n — rank(A2) = n — rank(A) = dim(null(A)) and, since null(A) Q null(A2), it follows that null(A) = null(A2). Suppose now that y belongs to null(A) n col(A). Then y = Ax for some at in R" and Ay = 0. Since A”): = Ay = 0, it follows that the vector x belongs to null(A2) = null(A), and so y = Ax = 0. This shows that null(A) ﬂ col(A) = {0}. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online