 4.6.1: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.2: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.3: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.4: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.5: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.6: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.7: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.8: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.9: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.10: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.11: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.12: In Exercises 112, find (a) the rank of the matrix, (b) a basis for ...
 4.6.13: In Exercises 1316, find a basis for the subspace of spanned by
 4.6.14: In Exercises 1316, find a basis for the subspace of spanned by
 4.6.15: In Exercises 1316, find a basis for the subspace of spanned by
 4.6.16: In Exercises 1316, find a basis for the subspace of spanned by
 4.6.17: In Exercises 1720, find a basis for the subspace of spanned by S
 4.6.18: In Exercises 1720, find a basis for the subspace of spanned by S
 4.6.19: In Exercises 1720, find a basis for the subspace of spanned by S
 4.6.20: In Exercises 1720, find a basis for the subspace of spanned by S
 4.6.21: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.22: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.23: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.24: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.25: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.26: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.27: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.28: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.29: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.30: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.31: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.32: In Exercises 2132, find a basis for, and the dimension of, the solu...
 4.6.33: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.34: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.35: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.36: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.37: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.38: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.39: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.40: In Exercises 3340, find (a) a basis for and (b) the dimension of th...
 4.6.41: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.42: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.43: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.44: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.45: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.46: In Exercises 4146, (a) determine whether the nonhomogeneous system ...
 4.6.47: In Exercises 4750, determine whether is in the column space of If i...
 4.6.48: In Exercises 4750, determine whether is in the column space of If i...
 4.6.49: In Exercises 4750, determine whether is in the column space of If i...
 4.6.50: In Exercises 4750, determine whether is in the column space of If i...
 4.6.51: Writing Explain why the row vectors of a matrix form a linearly dep...
 4.6.52: Writing Explain why the column vectors of a matrix form a linearly ...
 4.6.53: Prove that if is not square, then either the row vectors of or the ...
 4.6.54: Give an example showing that the rank of the product of two matrice...
 4.6.55: Give examples of matrices and of the same size such that (a) and (b...
 4.6.56: Prove that the nonzero row vectors of a matrix in rowechelon form ...
 4.6.57: Let be an matrix (where ) whose rank is (a) What is the largest val...
 4.6.58: Show that the three points and in a plane are collinear if and only...
 4.6.59: Given matrices and show that the row vectors of are in the row spac...
 4.6.60: Find the ranks of the matrix for and 4. Can you find a pattern in t...
 4.6.61: Prove each property of the system of linear equations in variables ...
 4.6.62: (a) The nullspace of is also called the solution space of (b) The n...
 4.6.63: (a) If an matrix is rowequivalent to an matrix then the row space ...
 4.6.64: (a) If an matrix can be obtained from elementary row operations on ...
 4.6.65: (a) The column space of a matrix is equal to the row space of (b) R...
 4.6.66: In Exercises 66 and 67, use the fact that matrices and are rowequi...
 4.6.67: In Exercises 66 and 67, use the fact that matrices and are rowequi...
 4.6.68: Let be an matrix. (a) Prove that the system of linear equations is ...
 4.6.69: Let and be square matrices of order satisfying for all (a) Find the...
 4.6.70: Let be an matrix. Prove that
 4.6.71: Prove that row operations do not change the dependency relationship...
Solutions for Chapter 4.6: Rank of a Matrix and Systems of Linear Equations
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 4.6: Rank of a Matrix and Systems of Linear Equations
Get Full SolutionsChapter 4.6: Rank of a Matrix and Systems of Linear Equations includes 71 full stepbystep solutions. Since 71 problems in chapter 4.6: Rank of a Matrix and Systems of Linear Equations have been answered, more than 18655 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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