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# Solutions for Chapter 5.2: Systems of Linear Equations in Three Variables

## Full solutions for College Algebra | 6th Edition

ISBN: 9780321782281

Solutions for Chapter 5.2: Systems of Linear Equations in Three Variables

Solutions for Chapter 5.2
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##### ISBN: 9780321782281

This textbook survival guide was created for the textbook: College Algebra , edition: 6. Since 57 problems in chapter 5.2: Systems of Linear Equations in Three Variables have been answered, more than 37286 students have viewed full step-by-step solutions from this chapter. Chapter 5.2: Systems of Linear Equations in Three Variables includes 57 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321782281.

Key Math Terms and definitions covered in this textbook
• 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).

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.

Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• Norm

IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

• Normal matrix.

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

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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