Linear Algebra: Matrices Flashcards

1
Q

Linear System

A

A finite set of linear equations in the variables x1, x2, … xn

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2
Q

Inconsistent linear system

A

A system with no solutions.

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3
Q

Consistent linear system

A

A system with at least one solution.

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4
Q

3 Elementary row operations

A
  • Multiply a row throughout by a nonzero constant - Interchange two rows - Add a multiple of one row to another
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5
Q

Row Echelon form

A

A matrix with properties: - The first nonzero number in every row is a 1 (leading 1). (Except if a row consists entirely of zeros). - Any rows consisting entirely of zeros are grouped together at the bottom of the matrix. - The leading 1 in the lower row occurs further to the right than the leading row in the higher row (in non-zero rows).

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6
Q

Reduced Row Echelon form

A

A matrix with all the properties of Row Echelon form and: - Each column that has a leading 1 has zeros everywhere else in that column.

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7
Q

General Solution

A

A set of parametric equations from which all solutions of a linear system can be obtained by assigning numerical values to the parameters.

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8
Q

Homogeneous

A

A system of linear equations where the constant terms are all zero.

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9
Q

Trivial solution to a homogeneous system

A

x1 = x2 = …. = xn = 0

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10
Q

Nontrivial solution to a homogeneous system

A

Any solution other than the trivial solution

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11
Q

2 Possibilities for any homogenous linear system:

A
  • Only the trivial solution - Infinitely many solutions, including the trivial solution.
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12
Q

Number of solutions to a homogenous system with more unknowns than equations.

A

Infinite

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13
Q

Number of free variables in a homogenous linear system that has n unknowns, and a reduced row echelon form with r nonzero rows.

A

n-r

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14
Q

Pivot positions

A

The positions of the leading 1’s.

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15
Q

Matrix

A

A rectangular array of numbers - called entries.

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16
Q

Column matrix

A

A matrix with only one column

17
Q

Row matrix

A

A matrix with only one row

18
Q

Scalars

A

Numerical quantities

19
Q

Equal matrices

A

Matrices with the same size, whose corresponding entries are equal.

20
Q

Transpose of a matrix

A

A matrix of the same size as the original, obtained by interchanging the rows and columns of A.

21
Q

Trace

A

The trace of a square matrix, denoted tr(A), is the sum of the entries on the main diagonal of the matrix.

22
Q

Zero matrix

A

Any matrix with only zero entries.

23
Q

Identity matrix

A

Square matrix with 1’s on the main diagonal and zero’s elsewhere.

24
Q

Explain an inverse

A

If A is a square matrix, and B a matrix of the same size, such that AB = BA = I, then A is said to be invertible and B is called an inverse of A.

25
Singular matrix
A matrix that has no inverse.
26
Nonnegative integer powers of a matrix (A)
A0 = I An = AA ... A (n times, n\>0)
27
Negative integer powers of a matrix (A)
A-n = ( A-1 )n = A-1 A-1... A-1 (n times)
28
Exponential Laws of matrices
If A is a square matrix and r and s are integers then: * ArAs = Ar+s * ( A)s = Ars
29
Define constant a in matrix terms
a.I
30
5 Properties of the transpose of a matrix
* ( (A))T = A * ( A+B )T = AT + BT * ( A-B )T = AT - BT * ( kA )T = kA, k being any scalar * ( AB )T = BTAT
31
Elementary matrix
A matrix that can be obtained from the nxn identity matrix In by performing a single elementary row operation.
32
Theorem 1.5.3 on matrices (4 all true/all false)
* A is invertible * Ax = **0** has only the trivial solution * The reduced row echelon form of A is In * A is expressible as a product of elementary matrices
33
Row equivalent matrices
Matrices that can be obtained from one another by a finite sequence of elementary row operations.
34
Method for inverting matrices
We must find a sequence of elementary row operations that reduces A to the identity and then perofrm this same sequence of operations to In to obtain A-1.