 4.5.1: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.2: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.3: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.4: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.5: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.6: In Exercises 16, find a basis for the solution space of the homogen...
 4.5.7: In each part, find a basis for the given subspace of R3, and state ...
 4.5.8: In each part, find a basis for the given subspace of R4, and state ...
 4.5.9: Find the dimension of each of the following vector spaces. (a) The ...
 4.5.10: Find the dimension of the subspace of P3 consisting of all polynomi...
 4.5.11: a) Show that the set W of all polynomials in P2 such that p(1) = 0 ...
 4.5.12: Find a standard basis vector for R3 that can be added to the set {v...
 4.5.13: Find standard basis vectors for R4 that can be added to the set {v1...
 4.5.14: Let {v1, v2, v3} be a basis for a vector space V. Show that {u1, u2...
 4.5.15: The vectors v1 = (1, 2, 3) and v2 = (0, 5, 3) are linearly independ...
 4.5.16: The vectors v1 = (1, 0, 0, 0) and v2 = (1, 1, 0, 0) are linearly in...
 4.5.17: Find a basis for the subspace of R3 that is spanned by the vectors ...
 4.5.18: Find a basis for the subspace of R4 that is spanned by the vectors ...
 4.5.19: In each part, let TA: R3 R3 be multiplication by A and find the dim...
 4.5.20: In each part, let TA be multiplication by A and find the dimension ...
 4.5.21: (a) Prove that for every positive integer n, one can find n + 1 lin...
 4.5.22: (a) Prove that for every positive integer n, one can find n + 1 lin...
 4.5.23: Let S = {v1, v2,..., vr} be a nonempty set of vectors in an ndimen...
 4.5.24: Prove part (a) of Theorem 4.5.6.
 4.5.25: Prove: A subspace of a finitedimensional vector space is finitedi...
 4.5.26: State the two parts of Theorem 4.5.2 in contrapositive form
 4.5.27: In each part, let S be the standard basis for P2. Use the results p...
Solutions for Chapter 4.5: Dimension
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 4.5: Dimension
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 27 problems in chapter 4.5: Dimension have been answered, more than 16800 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Chapter 4.5: Dimension includes 27 full stepbystep solutions. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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

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.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.