 5.1.1: Find the length of each of the vectors v in Exercises 1 through 3.
 5.1.2: Find the length of each of the vectors v in Exercises 1 through 3.
 5.1.3: Find the length of each of the vectors v in Exercises 1 through 3.
 5.1.4: Find the angle $ between each of the pairs of vectors u and v in Ex...
 5.1.5: Find the angle $ between each of the pairs of vectors u and v in Ex...
 5.1.6: Find the angle $ between each of the pairs of vectors u and v in Ex...
 5.1.7: For each pair of vectors u, v listed in Exercises 1 through 9, dete...
 5.1.8: For each pair of vectors u, v listed in Exercises 1 through 9, dete...
 5.1.9: For each pair of vectors u, v listed in Exercises 1 through 9, dete...
 5.1.10: For which value(s) of the constant k are the vectorsand 55 =u =perp...
 5.1.11: Consider the vectors' f V 1 0 and 53 =_1_ _0_in Rn.a. For n = 2, 3,...
 5.1.12: Give an algebraic proof for the triangle inequalityII5 + u>\\ < IlS...
 5.1.13: Leg traction. The accompanying figure shows how a leg may be stretc...
 5.1.14: Leonardo da Vinci and the resolution of forces. Leonardo (14521519...
 5.1.15: Consider the vector4Find a basis of the subspace of R4 consisting o...
 5.1.16: Consider the vectors"l/2" 1/2' 1/2" 1/2 _ 1/2  1 / 2Ml =1/2. K 2 =...
 5.1.17: Find a basis for W1, where/ f 5" \ 2 6 3 7 V 4 8
 5.1.18: Here is an infinitedimensional version of Euclidean space: In the ...
 5.1.19: For a line L in R2, draw a sketch to interpret the following transf...
 5.1.20: Refer to Figure 13 of this section. The leastsquares line for thes...
 5.1.21: Find scalars a, b, c, d, e, /, g such that the vectorsa b c d , 1 ,...
 5.1.22: Consider a basis , S2,..., vm of a subspace V of Rw. Show that a ve...
 5.1.23: Prove Theorem 5.1.8d. (V*)1 = V for any subspace V of R". Hint: ...
 5.1.24: Complete the proof of Theorem 5.1.4: Orthogonal projections are lin...
 5.1.25: a. Consider a vector v in Rn, and a scalar k. Show that11*511 = I*I...
 5.1.26: Find the orthogonal projection of space of R3 spanned by
 5.1.27: Find the orthogonal projection of 9e\ onto the subspace of R4 spann...
 5.1.28: Find the orthogonal projection ofY 0 0 _ 0_onto the subspace of R4 ...
 5.1.29: Consider the orthonormal vectors u\, U2, 3, W4, U5 in R 10. Find th...
 5.1.30: Consider a subspace V of Rn and a vector x in R'7. Let " 3 5 11 y =...
 5.1.31: Consider the orthonormal vectors u 1, U 2........um in R", and an a...
 5.1.32: Consider two vectors and x>2 in R". Form the matrixG =v\ vi v2 51V]...
 5.1.33: Among all the vectors in R" whose components add up to 1, find the ...
 5.1.34: Among all the unit vectors in R", find the one for which the sum of...
 5.1.35: Among all the unit vectors u = in R ,find the onefor which the sum ...
 5.1.36: There are three exams in your linear algebra class, and you theoriz...
 5.1.37: Consider a plane V in R3 with orthonormal basis u 1, ui. Let x be a...
 5.1.38: Consider three unit vectors v\,i>2, and 1)3 in R'7. We are told tha...
 5.1.39: Can you find a line L in R'7 and a vector jc in Rn such that x proj...
 5.1.40: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.41: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.42: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.43: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.44: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.45: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
 5.1.46: In Exercises 40 through 46, consider vectors v\, V2, v$ in R4; we a...
Solutions for Chapter 5.1: Orthogonal Projections and Orthonormal Bases
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 5.1: Orthogonal Projections and Orthonormal Bases
Get Full SolutionsSince 46 problems in chapter 5.1: Orthogonal Projections and Orthonormal Bases have been answered, more than 14334 students have viewed full stepbystep solutions from this chapter. Chapter 5.1: Orthogonal Projections and Orthonormal Bases includes 46 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269.

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.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.