~t A = [ ~ ~ :] and u = G J Repre>ent Au as a linear

In Eurcises 1- 16, compwe the mmrix- vector products. [! ~ J[ -i]

In Eurcises 1- 16, compwe the mmrix- vector products. [jr [

In Eurcises 1- 16, compwe the mmrix- vector products. [! ~ J[ -i]

In Eurcises 1- 16, compwe the mmrix- vector products. [! ~ J[ -i]

In Eurcises 1- 16, compwe the mmrix- vector products. [jr [

In Eurcises 1- 16, compwe the mmrix- vector products. 6. [2 3] [ -:]

In Eurcises 1- 16, compwe the mmrix- vector products. [~ ~r [~J

In Eurcises 1- 16, compwe the mmrix- vector products. [~ ~ ~] ]

In Eurcises 1- 16, compwe the mmrix- vector products. [~ ~ ~] ]

In Eurcises 1- 16, compwe the mmrix- vector products. [jr [

In Eurcises 1- 16, compwe the mmrix- vector products. [ ~ ~~] [~]

In Eurcises 1- 16, compwe the mmrix- vector products. [ ~ -~] [~]

In Eurcises 1- 16, compwe the mmrix- vector products. [ -~ ~ -~J ~]

In Eurcises 1- 16, compwe the mmrix- vector products. [-~ - ; -~] [:]

In Eurcises 1- 16, compwe the mmrix- vector products. ([ -~ r + [~ _;f)[~J

In Eurcises 1- 16, compwe the mmrix- vector products. 16. ([ -~ ~J + [~ -~D [~J

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (/ = 45. U = Cz

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (/ = 0. U = C1

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 60. II = m

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 30 . .. = [~J

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. fl=210 ~

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. fi= l35, u = [-7]

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. o = no, u = [ -;J

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. o = 330. u = [~]

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 240 . .. = [ =n

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 150 . .. = [ -~J

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 300 . .. = m

In Eurcises 17- 28, an angle 0 and a 1ector u are given. Write the corresponding rotation matrix, and compwe the vectorfowtd by rotating u by tire angle 0. Draw a sketch and simplify yortr answers. (! = 300 . .. = m

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [:].s= mJ.[~]}

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [ _:]. S = {[ _:]}

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [ _:]. S= {[:]}

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [:]. s = {[~]. [ -~Jl

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [i]. S = { [~l [ j]}

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. .. = [:J. s =I [~J. [-n [~J

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. . .. = [ : l s = w J. [-m

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. .. = [:l S= mJ. [ -n[:J

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = m. S= {[~].[;].[=;]}

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. U= ~J.s= {[:J.[-m

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u= [_Hs= {[j].[-l

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. = [ -n S = { [!J. [ -iJ

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. II= [-Us= [-!].[-~ -:] }

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. u = [H S= { GJ. [!] '[~ }

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. = [ =n 5 = { [~], [~ , [~ }

In Exercises 29-44, a vector u and a setS are given. If possible. write u as a linear combination of the \'ectors in S. . = [ -n S= { [-:J. . [ -iJ

A linear combina1ion of vectors is a sum of scalar mulliples of the vectors.

The coefficients in a linear combinalion can always be chosen to be positive scalars.

Every vector in n2 can be wriuen as a linear combination of the standard vectors of n 2

E,ery vector in n2 i; a linear combinalion of any two nonparallel vectors.

The zero vector is a linear combination of any nonempty set of vectors

The matrix-vector product of a 2 x 3 matrix and a 3 x 1 vector is a 3 x 1 vector.

The matrix vector product of a 2 x 3 matrix and a 3 x I vector equals a linear combination of the rows of the matrix.

The product of a matrix and a standard ,ector equals a standard vector.

The rotation matrix A 0 equals - / 2

The matrix vector product of an m x 11 matrix and " vector yield' a vector in R.".

Every vector in n2 i> a linear combination of two parallel vectors.

Every vector v in n" can be wriuen as a linear combination of the standard vectors. using the components of v as the coefficients of the linear combination.

A vector with exactly one nonzero component is called a standard vector.

If A is an m x 11 matrix, u ~> a vector in n". and c is a scalar, then A(cu) = c(Au)

If A is an m x 11 matrix. then the only vector u in n such that Au = 0 is u = 0.

For any \'ector u in n2 u is the vector obtained by rotating u by the angle 0.

If 0 > 0. then A. u i> the vector obtained by rotating u by a clockwise rotation of the angle().

If A i' iln m x 11 matri' and u and ,. are vectors in n such that Au = A' ' then u = v.

The matrix vector product of an m x 11 matrix A and a \"eCtor U in n equaJ, 11(81 + 112 3 1 - + 11n 3 n.

A matnx havmg nonncgauve entries such that the sum of the entrie' in each column is I is called a stochastic matrix.

Use a matrix vector product to >how that if () = 0. then Aov = v for all v in n2

Usc a matrix vector product to show that if 0 = 180. then Aov = - v for :til v in n2

Usc matrix- vector products to >how that. for any angles 0 and fJ and any vector v in n2 Ao(A,9 v) = A0+.8"

Compute ~(Ao ) and (A~ u) for any vector u in n2 and any :mgle 0.

Suppose that in a metropolitan area there me 400 thousand people living in the city and 300 thousand people living in the ;uburbs. Use the >tochastic matrix in Example 3 to determine (a) the number of people living in the city and suburbs after one year; (b) the number of people living in the city and suburbs after two years.

~t A = [ ~ ~ :] and u = G J Repre>ent Au as a linear combination of the columns of A. hr rercises 71- 74,1t't A = [ -~ ~] a11d u = [~ l

Show that Au is the reRcction of u about the yaxis

Prove that A(Au) = u.

Modify the matrix A to obtain a matrix 8 ..a that Bu i> the reRection of u about the ,, -axis.

~t C denote the rotation matrix that corre,pond> to () = 180. (a) Find C. (b) Use the matrix B in Exercise 73 to >how that A(Cu) = C(Au) = Bu and B(Cu) = C(Bu) = Au. (c) Interpret these equations in terms of renections and rotations. /11 Erercises 75-79. let A = [ ~ ~] and u = (;: l

Show that Au is the projection of tt on the xaxis.

Prove that A(Au) = Au.

Show that if v is any vector whose endpoint lies on the x-axis. then Av = ' '

Modify the matrix A to obtain a matrix 8 so that Bu is the proJection of u on the \'-axi>.

~t C denote the rotation matrix that corresponds to 8 = 180 . (See Exerme 74(a).) (a) Prove that A(Cu) = C(Au). (b) Interpret the result 111 (a) geometncally.

~t u1 and u2 be vectors rn n. Pro\"e that the >urn of two linear combination; of these vectors is also a tinear combination of the>e vecton..

~t 11 1 and 112 be vecton. in n. Let v and w be linear combination' of u 1 and u2. Prove that any linear combination of v and w is also a linear combination of u 1 and u2

~t u1 and u 2 be vectors in n. Prove that a >c:llar multiple of a linear combinatio n of thc>c vectors is also a linear combination of these vectors.

Prove (b) of Theorem 1.3.

Prove (c) of Theorem 1.3.

Prove (d) of Theorem 1.3.

Prove (e) of Theotcm 1.3.

Prove (f) of Theorem 1.3.

Prove (g) of Theorem 1.3.

Prove (h) of Theorem 1.3.

In reference to Exercise 69. dete rmine the number of pcopie living in the city and suburbs after 10 years.

For the matrices ['' 1.3 -0.1 ''] A= 1.3 -9.9 4.5 6.2 4.4 -2.2 5.7 2.0 0.2 9.8 1.1 -8.5 and [ '' 1.1 3.0 ''] B = -1.2 4.8 2.4 6.0 1.3 2.4 -5.8 2.8 6.0 -2.1 -5.3 8.2 1.3 Systems of Linear Equations 27 and the vectors "{l] (a) compute Au: (b) compute B(u + v); (c) compute (A + B )v: (d) compute A(Bv).