Solved: !11 Exerci;es 1- 12. find a genemting set for the

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T: 'R.2 4 R.2 d\

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T: 'R.2 4 R.2 d\

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. : n2 -+ n3 defined by T ([ ]) =

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T.: n2 -+ n3 defined by T ([ ]) = 1545

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. : n2 -+ n3 defined by T ([ ]) = 2x1464

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. ~: n[~ 4J n: d[e~;:~ ~

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. ~: n[~ 4J n: d[e~;:~ ~

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T 1<' ~ 1<' dofi.W

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. n3 4 n 3 defined by T [:~

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T: n3 4 R.3 defined by T(v) = v for all v in n 3

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T: nJ 4 R.2 defined by T(v) = 0 f

!11 Exerci;es 1- 12. find a genemting set for the range of each linear tramformmion T. T: R.3 4 n3 defined by T(v) = 4v for al l v in n

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R2 --> 'R2 defined by T

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R2 --> 'R2 defi ned by T

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R2 --> 'R2 defi ned by T

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R2 --> 'R2 defi ned by T

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R2 --> 'R2 defi ned by T

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. R3 --> 'R3 defined by

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. R3 --> 'R3 defined by

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. . T: 'R3 -+ 'R2 defined by

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: n2 .... n2 defined by r ([:~~ ]) =

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R3

In Exercises 13- 23, find tt genertlling set for 1he null space of each linear trall.iformatioll T, a11d use your a11swer To deTermine whether T is one-to-one. T: 'R4 ---. 'R3 defi ned by

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. 2 --. n2 defi ned by T ([~~]) = [x,

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. 'R2 --> 'R2 defi ned by T ([x'

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. T: RRR

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. T:R1R2R3

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. T:R3 R4

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. T:R3 R4

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. R 2x1 + 3xz ] X3 X i + X2 + 4X3

In rercises 24- 3 /,find The swndard mmrix of each linear 1/'tmsfomuuion T, tmd use i110 delennine wherher T is one-To-one. R 2x1 + 3xz ] X3 X i + X2 + 4X3

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. T: n z-+ n 2 defined by T

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. T: 'R2 __,. 'R2 defined by

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. T : nl .... n2 defined by T

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. r: n 2 .... n3 defined by T

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. T : 'R3 --> 'R defi ned by T (

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo.

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. l n d[efi:~d b~ + 2, 3 ] T ~2 = -2<, +x2 - 7XJ XJ X i - X2 + 2<3 X4 -x,+

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo.

In Exercises 32-40,find the sTandard matrix of each linear Transforma/ion T, and use it to de/ermine wheTher T is olllo. n[ ~l n d[e~ ~~ ~; + 2r3 +xl T X2 = x, + U 2 +X.! XJ Xi + X2 +X) + 2q x 3x, + 2rz + Sx4

A linear transformation with codomain 'R"' is onto if and only if the rank of its standard matrix eqmtls

A linear transformation is onto if and o nly if the columns of its standard matrix form a generating set for its range.

A linear transform

A linear transfo rmation is one-to-one if and only if every vector in its range is the image of a unique vector in its domain.

A linear transformation is one-to-one if and only if its null space consists only of the zero vector.

A linear transformation is invertible if and only if its standard matrix is invertible.

The system Ax = b is consistent for all b if and only if the transfom1ation TA is one-to-one.

Let A be an m x 11 matrix. The system Ax = b is consistent for all b in n"' if

A function is onto if its mnge equals its domain.

A function is onto if its mnge equals its codomain.

The set {T(et). T(e2) . .. . , T(e,.) } is a generating set for the range of any function T : n- n"'.

A linear transformation T: n" - n"' is onto if and only if the rank of its standard matrix is 11 .

The null space of a linear transformation T: n -> nm is the set of vectors in n" whose image is 0.

A function T: n -> nm is one-to-one if the only vector v in n" whose image is 0 is v = 0.

A linear transformation T: n" -> n"' is one-to-one if and only if the rank of its standard matrix is 111 .

If the composition UT of two linear transformations T: n"-+ nm and U: nP-+ nq is defined.thcnm = p.

The composition of linear transformations is a linear transformation.

If T: n" - nm and U : 'RJ' - n" are linear transformations with standard matrices A and 8. respectively. the standard matrix of TU equals BA.

For every invertible linear transformation T. the function r -l is a linear transformation.

If A is the standard matrix of an invertible linear transformation T, then the standard matrix of r- 1 is A - 1

Suppose that T: n2 -+ n2 is the linear transformation that rotates a vector by 90. (a) What is the null space of T? (b) Is T one-to-one? (c) What is the range ofT? (d) Is T onto?

Suppose that T: n2 -+ n2 is the reflection of n2 about the x-axis. (See Exerc ise 73 of Section 1.2.) (a) What is the null space of T? (b) Is Tone-to-one? (~) What is the range of T? (d) ls T onto?

Define T : n2 -+ n2 by T ([::~]) = [~l which is the projection of [ ~] on they-axis. (a) What is the null space of T ? (b) Is T one-toone? (c) What is the range ofT? (d) Is T onto?

Define T: n3 ..... n3 by T ([~:]) the projection of [; ] o n the z -axis. X) (a) What is the null space of T? (b) Is T one-toone'l (c) What is the range of T? (d) Is T onto?

Define T: n3 -. n3 by T [;~ ]) [;~ J which is the projection of [;:] o n the ,ty-plane. XJ (a) What is the null space of T ? (b) Is T one-to-one? (c) What is the range of T? (d) Is T onto?

efine T: n3 ~ n 3 by T ([;~ ]) [ _::] (See Exercise 92 in Section 2.7.) (a) What is the null space of T? (b) Is T one-toone? (c) What is the

Suppose that T: n2 -> n2 is linear and has the property that T(et) = m

Suppose that T: n 2 -> n2 is linear and has the property that T(e 1) = [~] and T(e2) = [a (a) Determine whether T is one-toone. (b) Determine whether T is onto.

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Dclermine I he domain, I he codom:.tin, and lhe rule for UT.

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Uc the rule for UT obtained in Exercise 69 10 lind lhe >l:tndard malrix of UT

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Dclermine the standard matrices A and 8 of T and U. repet1ively.

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Compute the produc1 BA of 1he matrices found in Exercise 7

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Delermine the domain. the codomain. and the rule for TU.

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Ue the rule for TU ob1ained in Exercise 73 to find the Mandard matrix of TU .

Exercises 69-75 are co11cemed with the li11ear tra11sj'ormations T: n2 - n3 and U: n3 -+ n2 defined as Compute the product AB of lhc malrices found in xercise 7 1. and illustrale Theorem 2. 12 by comparing your answer with lhe esuh oblained in Exercise 74.

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] De1ermine lhe domain. the codomain. and the rule for UT.

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] Ue 1he rule for UT ob1ained in Exerci'e 76 to find 1he Mandard matrix of UT.

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] De1em1ine 1he standard matrice' A and 8 of T and U. especlively

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] Compu1c the producl BA of 1he m:urices found in Exercise 78. and illuma1e Theorem 2. 12 by comparing your answer with the resuh obtained in Exercise

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] Detennine the domain, lhe codom:tin. and lhe rule for TU

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] U>e lhe rule for TU obwincd in Exercise 80 10 find the Mandard matrix of TU.

Exerdses 76 82 are concemed ll'itlr tire linear transformations T : n2 -+ n2 and U: n2 ~ n2 defined as T ([:~ ]) = [ :~: = ~~] and u ([ :~]) = [2x~~ Xl] Compute lhe product AB of the matrices found in Exercise 78. and illustrate Theorem 2.12 by comparing your answer with the resuh obtatned m Exerctse 8 I.

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation. T: n2 ~ n2 defined by T ([x']) = [2.rt -xz] 12 Xt +x

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation. T: n2 --+ n2 defined by T ([r

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation. n[~::] )n: d[e~~~~Y

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation.

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation.

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation.

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation.

In Exurius 83-90. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fined. ~t~ mrine a similar definition for tire imerse r-t of em/t lmeur trcmsfonuation. [(] n~ ['~~:;.~~;~~~l:] x, 13xa - 5xz + Sx3- 6x, 90.

Prove lhm the composition of two one-to-one linear trans formation is one-to-one. ls the result true if I he lrunsfor rnations are not linear? Justify your answer.

Prove th;rl if 1wo linear transformations are onto. then I heir composition is onto. Is the result true if the lmnsformations are not linear? Justify your answer.

In n2 sho" that the composition of two reflectton:. about the A -axis is the identity transformation.

In n2 show that a rellcction about the y-axi\ follo\\ed by :t rotation by 180 is equal to a reflection about the x-axis

In n2 show that the composition of the proJection on the .\-axis followed by the reflection about lhc y-axis is equal to the composition of the reflection about the )'axis followed by the projeclion on the x-axis.

Prove that the composition of two shear transforrmuions is a shear transfommtion. (See Example 4 of Secl

Suppose that T : n -+ n'" is linear and one-lo-one. Let (v,, ' '! .... " I he a linearly independent >Ub,ct of n. (a) Prove that the set (T(v 1). T(\ 2) ... T( v1 >1 is a lin early mdependent subset of n. (b) Show

Use Theorem 2.12 to prove that matrix multiplication is associative.

The linear tran;,formalions T. U: R 4 --+ n4 are defined

Define the linear transformation T: R.4 -> R.4 by the rule (a) Find the standard matrix A ofT. (b) Show that A is in vertible and find its in verse. (c) Use your answer to (b) to fi nd the rule for r