 4.1.1: Sketch a directed line segmenl in Rl. representing each of the foll...
 4.1.2: Determine the head of the vector [ ~] whose tail is (3. 2). Make a...
 4.1.3: Determine the tail of the vector [!] whose head is (t. 2). '1ake a...
 4.1.4: D""ml", .h, "I] of .he ""m [ _ n who~ heed I,
 4.1.5: For what values of 1.1 and b are the vectors [1.1 ~ b] and [1.I~b] ...
 4.1.6: Foe wh" ,,]"" of a. b. "d ,,,"h,,~,",, [;'~ ~] "d [ ~ ] 'q"'"
 4.1.7: III Nercises 7 alldS. determille thecompollent.Yoj each l'eCfOr po.
 4.1.8: III Nercises 7 alldS. determille thecompollent.Yoj each l'eCfOr po.
 4.1.9: In Erercil'es 9 {llId 10. filld .!!...:!'ector ",hose wil i.I' lire...
 4.1.10: In Erercil'es 9 {llId 10. filld .!!...:!'ector ",hose wil i.I' lire...
 4.1.11: Compute u + v. u  v. 2u. and 3u  2v if (, ) ~[;] , ~ [;J (h, ~n ...
 4.1.12: Compute u + v. 2u  v. 3u  2v. and O 3v if (a) u = m ,~ m (b) u =...
 4.1.13: Let c = 2, and d = 3. Compllte each of the following: I' i H' (h, ...
 4.1.14: Let ~[;l Y~ !l ~[:l ~d ~[ ;l Find rand s so that (a) z = 2x. =y. (e...
 4.1.15: Let Find r .. \'. and I so that (a) z=tx. (b) z+ u = x. (e) z  x =y.
 4.1.16: [fpossible. find scalars CI and C2 so that
 4.1.17: If possible. find scalars CI. C2. and c ) so that
 4.1.18: [f possible. find scalars CI and q . not both zero. so that
 4.1.19: [fpossible. find scalars cI.q.and CJ. not all zero. so that
 4.1.20: Let F;od ",'," ,.,. ,.,. eod " '0 ,h" ,oy ",",0' " ~ [;]
 4.1.21: Show that if u isa vector in R!or R). then u + O= u.
 4.1.22: Show that if u is a vector in R2 or RJ. then u + (I)u=O.
 4.1.23: Prove part (b) and parts (d) through (h) of Theorem 4.1 .
 4.1.24: Determine whether the software you use supports graphics. If it doe...
 4.1.25: Assuming that the software you lise SlIpportS graphics (see Exercis...
 4.1.26: Determine whether the software you use supports threedimensional gr...
Solutions for Chapter 4.1: Vectors in the Plane and in 3Spoce
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 4.1: Vectors in the Plane and in 3Spoce
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Chapter 4.1: Vectors in the Plane and in 3Spoce includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 26 problems in chapter 4.1: Vectors in the Plane and in 3Spoce have been answered, more than 9415 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.