Math 340 Spring 08Lecture 4 G.MeyerPractice 21. LetA =1 1 2 31 2 3 10 −1 −1 2.a) Find the dimension and a basis for the null space of A.b) Find dimension and a basis for the row space of A consisting of(i) not necessarily row vectors of A(ii) row vectors of Ac) Find the dimension and a basis for the column space of A con-sisting of(i) not necessarily column vectors of A(i) column vectors of Ad) What is the rank of A?e) Find the general solution to Ax = b, where b =220, i.e.x = g + h, where g is a solution to Ax = b and h is a solution forAx = 0.2. Let V be the subspace of R4spanned by the vectors1001,0110, and1111What is the dimension of V ?Find a basis for V .13. Let S = {1−10,010,102} and T = {6−52,110,012} bebases for R3.a) Find the transition matrix PS←T.b) The coordinates of the vector v with respect to the standardbasis is−413. Find [v]S.c) Given [v]T=−220, find [v].4. Answer the following questions as True (T) or False (F) by circling Tor F below. No justification wanted.(T) (F) There are five linearly independent vectors in R4.(T) (F) I f v1, v2, v3span a 3-dimensional subspace V of R4, the nthey are a basis of V .(T) (F) If A is a 8 by 6 matrix, then you can be sure that thehomogeneous system Ax = 0 has a nontrivial solution.(T) (F) A linearly independent set of 4 vectors in a 7-dimensionalvector space V can always be enlarged to a basis of V .(T) (F) The set of all nonsingular 3 by 3 matrices is a subspace ofthe vector space M3,3of all 3 by 3 matrices.(T) (F) A 2 by 6 matrix could have rank 4.(T) (F) If the nullity (dimension of nullspace) of the 4 by 4 matrixA is 0, then A is nonsingular.(T) (F) The set of all real polynomials of degree ≤ 6, using stan-dard addition and scalar multiplication, is a vector space of
View Full Document
Unlocking...