Let L be the line in R3 spanned by the vectorLet T from R3 to R3 be the rotation about

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 1 through 18, determine whether the vectoi x is in the span V of the vectors v \, . . . , vm (proceed b) inspection if possible, and use the reduced row-echeloi form if necessary). If x is in V, find the coordinates of i with respect to the basis $3=( v i , . . . , v m)ofV, and writi the coordinate vector [jc]

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 19 through 24, find the matrix B of the linear transformation T(x) = Ax with respect to the basis 33 = (jTj, v2)- For practice, solve each problem in three ways: (a) Use the formula B = S~lAS (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B column by column.

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

In Exercises 25 through 30, find the matrix B of the linear transformation with respect to the basis

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

Let 93 = (vi, V 2, v3) be any basis o/M3 consisting of perpendicular unit vectors, suc/i that v$ = vx x v2-I*1 Exercises 31 through 36,find the 23-matrix B of the given linear transformation T from R3 to R3. Interpret T geometrically.

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal.Orthogonal projection T onto the line in R2 spanned by Y

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal.Reflection T about the line in R2 spanned by

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal.Reflection T about the line in R3 spanned by

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal.Orthogonal projection T onto the line in R3 spanned by 1' 1 _1

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal. Orthogonal projection T onto the plane 3x\ + x2 + 2x3 = 0 in R3

In Exercises 37 through 42, find a basis 23 of R" such that the 93-matrix B of the given linear transformation T is diagonal.Reflection T about the plane x\ 2x2 + 2*3 = 0 in R3

Consider the plane x\ + 2x2 + *3 = 0 with basis 93 con sisting of vectors find jc.

Consider the plane 2jci-3 jc2-1-4jc3 = 0 with basis 93 con5" and "-1" ~-2 0 and 1 if [*] = 2" -3 1 0 sisting of vectors find x.

Consider the plane 2x\ - 3*2 + 4; of this plane such that [jc

Consider the plane of this plane such that

Consider a linear transformation T from R2 to R2. We ion T from R2 to R2. We 1 -1 1 Y 'o' Y are Ci = 1 . h = 1 * c3 = 2 o l 2 4 1 is a b c d Find the standard matrix of T in terms of a, b, c, and d.

In the accompanying figure, sketch the vector x with -1 2 of the vectors C, w.

Consider the vectors 5, C, and w sketched in the accompanying figure. Find the coordinate vector of w with1 respect to the basis w, C.

Given a hexagonal tiling of the plane, such as you migtt find on a kitchen floor, consider the basis 93 of R2 coil: sisting of the vectors C, w in the following sketch:a. Find the coordinate vectors n P L J* L Hint: Sketch the coordinate grid defined by the basis 93 = (v, w). b. We are told that = 2 R. Is R a vertex or a center of a tile? 17' . Sketch the point Is S a center or ac. We are told that ^ vertex of a tile?

Prove part (a) of Theorem 3.4.2.

If 93 is a basis of Rn, is the transformation T from to R" given by T(x) = [i] linear? Justify your answer.

Consider the basis 23 of R2 consisting of the vectors and We are told that [jc] for a certain ^ = vector Jc in R2. Find j c.

Let 23 be the basis of Rn consisting of the vectors 5i, 02> 1 vn, and let X be some other basis of R". Is [5> ]f N j a basis of W1 as well? Explain.

Consider the basis 93 of R2 consisting of the vectors and , and let SK be the basis consisting of Find a matrix P such that [*]; = P [*] for all x in R2.

Find a basis 23 of R2 such that

Show that if a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix

Considej a 3 x 3 matrix A and a vector v in R3 such that A3v = 0, but A2v 0. a. Show that the vectors A2v, Av, v form a basis of R3. (Hint: It suffices to show linear independence. Consider a relation c\A2v + C2Av + civ = 0 and multiply by A2 to show that C3 = 0.) k Find the matrix of the transformation T(Jc) = Ax with respect to the basis A2v, Avy 0.

Is matrix similar to matrix

Is matrix similar to matrix

Find a basis 23 of R2 such that the 23-matrix of the linear transformation T(x) = -5 -9 4 7 Jc is B = 1 1 0 1

Find a basis 23 of R2 such that the 23-matrix of the linear transformation T(x) = 1 2 4 3 jc is B = 5 0 0 -1

Is matrix P -q similar to matrix P <1 3 P_ P _ p and ql

Is matrix a ^ similar to matrix c a a, b, c, dl

Prove parts (a) and (b) of Theorem 3.4.6.

Consider a matrix A of the form A = a c b d a b b a for all for all , where a2 + b2 = 1 and a ^ 1. Find the matrix B of the linear transformation T (jc) = Ax with respect to the basis . Interpret the answer geometrically.

If c ^ 0, find the matrix of the linear transformation r a b c d T(x) = x with respect to basis Y a 0 c

Find an invertible 2 x 2 matrix S such that 1 2 3 4 0 b 1 d is of the form

If A is a 2 x 2 matrix such that A . See Exercise 67. Y '3' 2 '-2 and A 2 6 1 -1 show that A is similar to a diagonal matrix D. Find an invertible S such that S~] AS = D.

Is there a basis 23 of R2 such that 23-matrix B of the linear transformation 0 - 1 T(x) = 1 0 is upper triangular? (Hint: Think about the first column of B.)

Suppose that matrix A is similar to B, with B = S-1AS. a. Show that if jc is in ker(Z?), then Sx is in ker(A).b. Show that nullity(A) = nullity(B). Hint: If 0i, 02,..., vp is a basis of ker(fl), then the vectors 5Ci, Sv2.......Svp in ker(A) are linearly independent. Now reverse the roles of A and B.

If A is similar to B, what is the relationship between rank(>4) and rank(fl)? See Exercise 71.

Let L be the line in R3 spanned by the vector Let T from R3 to R3 be the rotation about this line through an angle of 7t/2, in the direction indicated in the accompanying sketch. Find the matrix A such that T (Jc) = Ax.

Consider the regular tetrahedron in the accompanying sketch whose center is at the origin. Let 0o, 0i, 02, 03 be the position vectors of the four vertices of the tetrahedron: 00 = OPq,..., 03 = OP3. a. Find the sum 0o + 0i + 02 + 03b. Find the coordinate vector of 0o with respect to the basis 0i, 02, 03. c. Let T be the linear transformation with T(vo) = 03, T(v3) = 0i, and T(0i) = 0q. What is T(v2)? Describe the transformation T geometrically (as a reflection, rotation, projection, or whatever). Find the matrix B of T with respect to the basis 01, 02, 03. What is Z?3? Explain.

Find the matrix B of the rotation T(x) = "0.6 0" - f 0 = 0.8 with respect to the basis 1 0 0 swer geometrically.

If t is any real number, what is the matrix B of the linear transformation T(x) = cos (f) - sin(r) sin(r) cos(r) with respect to basis your answer geometrically.

Consider a linear transformation T(Jc) = Ax from Rn to R". Let B be the matrix of T with respect to the basis en, en-\,..., <?2, e\ ofR". Describe the entries of B in terms of the entries of A.

This problem refers to Leontiefs input-output model, first discussed in the Exercises 1.1.20 and 1.2.37. Consider three industries I\, I2, h , each of which produces only one good, with unit prices p\ = 2, P2 = 5, = 10 (in U.S. dollars), respectively. Let the three products be labeled good 1, good 2, and good 3. Let a\\ *12 013 "0.3 0.2 O.f A = 021 022 023 = 0.1 0.3 0.3 *31 032 033 0.2 0.2 0.1 be the matrix that lists the interindustry demand in terms of dollar amounts. The entry a/y tells us how many dollars worth of good / are required to produce one dollars worth of good j. Alternatively, the interindustry demand can be measured in units of goods by means of the matrix B = b 11 612 ^13 /?21 b22 b23 631 />32 />33 where bij tells us how many units of good / are require* to produce one unit of good j. Find the matrix B for the economy discussed here. Also write an equation relatim the three matrices A, By and 5, where 5 = 2 0 0 0 5 0 0 0 10 is the diagonal matrix listing the unit prices on the diagO nal. Justify your answer carefully.