 6.2.1: Let L : R2 ....... R" be the linea! transformation defined by (a ) ...
 6.2.2: Let L : R2 ;. R2 be the linea! operator defined by I' J Is [2 3  ...
 6.2.3: Let L : ~ ....... R2 be the linear tmnsfonnation defined by Ie) Fin...
 6.2.4: Let L : R2 + R3 be the linear transformation defined by L([1I 1 11...
 6.2.5: Let L : ~ + R3 be the linear trnnsfonnation defined by L([1I1 112 ...
 6.2.6: Let L: P2 + P3 be the linear trnnsfonmltion defined by L(p(l = 12,...
 6.2.7: Let L : M2j __ Mn be the lirrar transformation defined by 2 I] A I...
 6.2.8: Let L: 1'2 + PI be the linear transform.'1tion defined by L(tlt 2 ...
 6.2.9: LeI L:"2 > H.l be the linear trnllsformulion defined by L(a/ 2 +bl ...
 6.2.10: LeI L:"2 > H.l be the linear trnllsformulion defined by L(a/ 2 +bl ...
 6.2.11: lei L: M 22 + M n be the linear operator defined by ([a b]) [a +b ...
 6.2.12: leI L : V __ IV be a linear transfoml:ltion. (a) Show that dimrange...
 6.2.13: Verify Theorem 6.6 for the following linear transfonnalions: (a) L:...
 6.2.14: Verify Theorem 6.6 for the following linear transfonnalions: (a) L:...
 6.2.15: Lei A be an 11/ X " matrix. and consider the linear transfomlalion ...
 6.2.16: LeI L : RS + R' be the linear transformation defin~d by L([]){ 0 ...
 6.2.17: Let L : Rj ...... R3 be the linear tr.lIIsfonnation defined by L(e ...
 6.2.18: Let L : V ,I IV be a linear tmnsfonnation. and let dim V = dim I...
 6.2.19: Let L: 1(3 ,I 1(3 be defined by (a) Prove that L is invertible
 6.2.20: Let L : V __ IV be a linear transfonnation. and let 5 = {VI. V! ......
 6.2.21: Find the dimension of the solution space for the followmg homogeneo...
 6.2.22: Find a linear transfonnation L : 1(2 ,I R 3 such that S = ([I  ...
 6.2.23: Let L : f( l __ 1 be the Imeat transformation defined by
 6.2.24: Let L V + IV be a linear transfonnation. Prove that L is onetoon...
 6.2.25: Let L: R'; ___ 1(6 be a linear transfonnation. (II) If dim kef L = ...
 6.2.26: LeI L : V )0 RS be a linear transfonnation. (a) [f L is onto and ...
 6.2.27: Let L be the linear transformation defined in Exercise 24. Section ...
 6.2.28: Let L be the linear transformation defined in Exercise 25 . Section...
 6.2.29: Prove Corollary 6.1.
 6.2.30: Let L : RIO ..... Rm be a linear transformation defined by L(x) = A...
Solutions for Chapter 6.2: Kernel and Range of a linear Transformation
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 6.2: Kernel and Range of a linear Transformation
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 30 problems in chapter 6.2: Kernel and Range of a linear Transformation have been answered, more than 10191 students have viewed full stepbystep solutions from this chapter. Chapter 6.2: Kernel and Range of a linear Transformation includes 30 full stepbystep solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.