Matrices Flashcards
The range of a matrix is
the order of the largest square sub-matrix whose determinant is not 0
The range of A is written as
Rag A or rg(A).
The order of a square matrix is the
number of rows or columns in that matrix.
An array of order 2 is the same as saying ____________, which has ______ and _______.
an array of dimensions 2×2
2 rows
2 columns
A submatrix is
a matrix that is contained within another matrix
Within a matrix, we can choose rows and columns that form another independent matrix.
True
I can choose columns that are not sequential to form a submatrix.
True
I can choose fewer rows and columns to form a submatrix.
True
To calculate the range of a matrix, we must ____________.
choose the sub-matrix with the highest possible order and calculate its determinant
The _______ of the largest SQUARE sub-matrix, whose determinant is ________, will be the range of the matrix.
order
not zero
___________ matches the order of the largest square sub-matrix we have found, whose determinant is other than zero.
The range
Linearly independent vectors are ______________, i.e. which cannot be composed from the linear composition of the rest of the vectors.
the vectors whose formation does not depend on any other vector in the matrix
The linear vectors of an array is ______________.
the number of vectors other than zero that remain after having triangulated the array formed by them
The vectors of an array we are referring to are the _________, i.e. each row of the array corresponds to a vector.
row vectors
To obtain the number of linearly independent vectors, we are going to ____________, that is to say, we are going to make the ___________ zeros.
triangulate the matrix
elements below the main diagonal
The first element of the first column needs to be a 1, if we are going to triangulate the matrix.
True
If we have a 3x4 matrix with a row full of zeros, this matrix has ____________ vectors and as a consequence, the other vector is dependent on the other two.
two linearly independent
If the first element of a matrix is not a 1, we can exchange it for a vector in matrix that has 1 as its element.
True
When calculating the range from the Gaussian Method, if we have already triangulated a 3x4 matrix, and we are not left with any row with zeros, then the 3 vectors of the matrix are ___________. Therefore, as the matrix has 3 linearly independent vectors, its range is ___.
linearly independent
3
When triangulating a 3x3 matrix and we have 2 zeros in the first column and 1 zero in the second column, since the matrices had 3 rows, remember that to triangulate the matrix is to make ___________ and that in each case it will be different, depending on the number of rows in the matrix.
zeros below the main diagonal
In the Gaussian Method, the range of an array is the _________.
number of linearly independent vectors in the array
A system of two linear equations can have _________, an _____________, or _______. Systems of equations can be classified by ________.
one solution
infinite number of solutions
no solution
the number of solutions
If a system has at least one solution, it is said to be __________.
consistent
If a consistent system has exactly one solution, it is __________.
independent
If a consistent system has an infinite number of solutions, it is ______. When you graph the equations, both equations represent the same line.
dependent
If a system has no solution, it is said to be ____________. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.
inconsistent
“Independent” means that each equation gives new information.
Otherwise they are “Dependent”.
True
If the range of a 3x3 matrix is <3, the determinant is _____, and the matrix _________ an inverse, which means that it ___________ solution.
0
does not have
does not have a
No solution means the graph of two linear equations are ____________.
parallel
In order to find a general solution of a system of equations, one needs to simplify it as much as possible
True
The simplest system of linear equations is one where every equation has only one unknown and all these unknowns are different.
True
It is not possible to reduce every system of linear equations to this form, but we can get very close. There are three operations that one can apply to any system of linear equations: _______________
The system obtained after each of these operations is ____________ to the original system, meaning that they have the same solutions.
- Replace an equation by the sum of this equation and another equation multiplied by a number
- Swap two equations
- Multiply an equation by a non-zero number
equivalent
In order to simplify a system of equations, it is enough to simplify its _________ by using the following row operations:
augmented matrix
- Replace a row by this row plus another row multiplied by a number.
- Swap two rows.
- Multiply a row by a non-zero number.
We can get a general solution of every system of equations whose matrix is in the _________-.
reduced row echelon form
A system of linear equations either has no solutions or has exactly one solution or has infinitely many solutions
True
A system of linear equations has infinitely many solutions if and only if _______________.
its reduced row echelon form has free unknowns and the last column of the reduced row echelon form has no leading 1’s
A system of linear equations has exactly one solution if and only if _______________.
the reduced row echelon form has no free unknowns and the last column of the reduced row echelon form has no leading 1
A system of linear equations has no solutions if and only if the ______________.
last column of the reduced row echelon form has a leading 1
In row echelon form, _____________, _______________, and _________________.
- each leading entry must be 1
- each leading entry must lie to the right of the leading entry above it
- rows containing all zeros go last.
Let V be a subspace of R^n. The number of vectors in any basis of V is called the dimension of V, and is written dim V
True
A basis is in A is a set of vectors in A that:
- Span A
2. Are linearly independent
The dimension of A is the number of vectors in a ______________.
basis for A
In subset ℝ^3, ℋ = {(x,y,z)∈ ℝ3 ǀ x+ y = 0, z + 2y = 0}, “ x+ y = 0, z + 2y = 0” is _______________.
the rule of the basis
“Spans S” means that every vector in S is a scalar multiple of [x y … n].
True
The linear combination of a single vector is a ______________.
scalar multiple
The determinant of a triangular
matrix is the ________________ of its diagonal entries.
product
A square matrix A is invertible if and only if _________.
detA ≠ 0
det(AB) =
det(A) det(B)
det(A) det(A−1) = det(AA−1) = det(I)=_____.
1
If A is invertible, det(A^−1) = _________.
1/det A
If Ax = λx for some scalar λ and some nonzero vector x,
then we say ______________ and x is _________.
λ is an eigenvalue of A
an eigenvector associated with λ
Any (nonzero) scalar multiple of an eigenvector
is itself an eigenvector (associated w/same eigenvalue).
True
Eigenvalues of A means find values ____________.
of λ such that det(A − λI) = 0
Given eigenvalue λ, associated eigenvectors are
______________.
nonzero vectors in null(A − λI)
