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What does the term "inconsistent" in regards to linear systems?
The linear system has no solution.
What does the term "consistent" in regards to linear systems?
The linear system has at least one solution.
When are two systems of equations considered equivalent?
When they involve the same variable and have the same solution.
What are the three elementary row operations?

    • Interchange two rows.
    • Multiply a row by a non-zero real number.
    • Replace a row by its sum with a multiple of another row.

When is a system in strict triangular form?
If in the kth equation, the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1...n).
What is back substitution?
The process of solving an equation in strict triangular form by using xn to solve (n-1)st equation for xn-1 and using that value to solve the (n-2)nd equation for xn-2 and so on.
What is a coefficient matrix?
A matrix consisting of the coefficents of the variables on the left hand side of the linear equations.
What is an augmented matrix?
A coefficent matrix that includes the right hand side of the matrix as well.
What are lead variables?
Variables corresponding to the first nonzero elements in each row of a reduced matrix.
What are free variables?
Variables skipped in the reduction process to row eschelon form.
What are the three conditions for a matrix to be in row echelon form?

  • The first nonzero entry in each nonzero row is 1

  • If row k does not consist entirely of zeroes, the number of the leading zero entries in row k+1 is greater than the number of leading zero entries in row k.

  • If there are rows whose enteries are all zero, they are below the rows having nonzero entries.

What is Gaussian Elimination?
The process of using the three elementary row operations to transform a linear system into one whose augmented matrix is in row echelon form.
If a row of an augmented matrix contains the form [0 0 0 | 1], what does that mean for the linear system?
It is inconsistent.
When is a linear system overdetermined?
When it has more equations than unknowns.
Linear systems that are overdetemined are usually, but not always, what?
Inconsistent.
When is a linear system considered to be underdetermined?
When there are fewer equations than unknowns.
Linear systems that are undetermined usually have what kind of solution?
Undetermined systems are usually consistent with infinitely many solutions.
When is a matrix in reduced row echelon form?

  • The matrix is in row echelon form.

  • The first nonzero entry in each row is the only nonzero entry in its column.

What is Gauss-Jordan reduction?
The process of using elementary row operations to transform a matrix into reduced row echelon form.
When is a system of linear equations said to be homogeneous?
If the constants on the right hand side are all zero.
Homogeneous systems are always what?
Consistent.
An m X n homogeneous system of linear equations has a nontrivial solution when?
n > m
What is a linear combination?
If a1, a2,...an are vectors in Rm and c1,c2,...cn are scalars, then a sum of the form c1a1 + c2a2 + ... + cnan is the linear combination of the vectors a1,a2,...an.
What is a transpose of an m x n matrix named A.
the n x m matrix B defined by bij = aij for j = 1,...n and i = 1,...m
When is an n x n matrix A said to be symmetric?
If AT = A
What is the identity matrix?
An n x n matrix I = ijk where ijk ={ 1 if i = j or 0 if i ≠ j.
When is a matrix said to be nonsingular?
When an n x n matrix A has a paired matrix B such that AB = BA = I.
When is a matrix B a multiplicative inverse of matrix A?
When matrix B is an n x n matrix such that AB = BA = I.
When is a matrix singular?
When it is an n x n matrix that does not have a multiplicative inverse.
What type of matricies can be refered to as singular or nonsingular?
Square or n x n matricies only.
What is an elementary matrix?
An identity matrix that has been subject to exactly one elementary row operation.
When is a matrix B row equivalent to a matrix A?
If there exists a finite sequence of E1,E2,...Ek of elementary matrices such that B = EkEk-1...E1A
What are the three equivalent conditions for nonsingularity?

Let A be an n x n matrix, then the following are equivalent:
  • A is nonsingular
  • Ax = 0 has only the trivial solution 0.
  • A is row equivalent to I.
The system Ax = b of n linear equations in n unknowns has a unique solution iff?
A is nonsingular.
What is the consistency theorem for linear systems?
A linear system Ax = b is consistent iff b can be written as a linear combination of the column vectors of A.
The system Ax = b of n linear equations in n unknowns has a unique solution iff (corollary 1.5.3)?
A is nonsingular.
What is an inner product?  What is it also known as?
Given vectors and y they can be multiplied by transposing x, resulting in xTy.  This is also known as the scalar product.
What is an outer product?
Given the vectors and y, perform an operation similar to the inner product, but transpose instead resulting in xyT.
What is outer product expansion?
The concept of the outer product applied to matrices instead of  vectors, ie. for m x n matrix X and k x n matrix Y, a m x k matrix can be obtained by the following operation, XYT.
x of y cards