Prove or disprove
(8) Let T: V → V be linear. If T is diagonalizable, then T is invertible.
With V as in Exercise 11, define T: V-R2 by a) Prove that T is a linear transformation. b) Give an algebraic specification for N(T) c) Exhibit a basis for N(T). d) Determine the nullity and the rank of T
Q4
(b) Prove that ker P = ker Tn ker S. () h a Question 4. Define T: Ma2 = c+ d. Prove that T is a linear transformation R by T C and onto. Find dim(ker T). Is T one-to-one? Jamomials in P. Show
(b) Prove that ker P = ker Tn ker S. () h a Question 4. Define T: Ma2 = c+ d. Prove that T is a linear transformation R by T C and onto. Find...
Let ai, a2 , аз, bị, b2P3 R. Define T : R3 R2 by Prove T is a linear transformation.
Let
be a map
Define the map
prove or disprove
2)
for all
3)
for all
A B We were unable to transcribe this imagef(and) = f(c) n (D) CD CA f-1( EF) = f-1(E)f-1(F) We were unable to transcribe this image
a) Prove or disprove: if S,TELluv) then trace ('st) = trove's) tracel T) b) Prove or disprove, if S.TELIVE) then det (5+ 7) = det (5) + det (7)
both questions
-3 4 7. Prove or disprove, A: R3 R3 is bijective (1-1 and onto), where the standard matrix for A is A = -2 1 -1 3 8. Let A: R2 R2 be the linear transformation that that stretches the a-axis by a factor of 3, and the y-axis by a factor of 4. Find the standard matrix for A. 127
Prove or disprove the following equivalence claim. (r ∧ s ∨ ¬t) ⇒ q ≡ ( ¬r ∧ t) ∨ (¬s ∧ t) ∨ q
disprove the following statements (if it is true, please write a proof 1: (15 Points) Prove or or quote the corresponding theorem from the textbook; if it is false, please provide a counter example to disprove If u is orthogonal to all the vectors 1, U2,,n then u is orthogonal to all the vectors in Span({, ,., )
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...