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Linear Algebra Final Review
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Terms in this set (55)
what is a linear system?
Every system of linear equations has zero, one, or infinitely many solutions.
1. When lines are parallel and distinct there is no intersection and consequently no solution.
2. When lines intersect at only one point leaving only one solution.
3. When lines have infinitely many points of intersection and therefore infinitely many solutions
what is gaussian elimination?
(REF): Reduced Echelon Form
1. the first nonzero # in the row must be a one
2. zero rows are grouped together at the bottom of the matrix
3. if any two successive rows do not consists entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row
what is gaussian jordan elimination?
(RREF): Reduced Row Echelon Form
1. the first nonzero # in the row must be a one
2. zero rows are grouped together at the bottom of the matrix
3. if any two successive rows do not consists entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row
4. Each column that contains a leading 1 has zeros everywhere else in that colum
what is an augmented matrix?
this may be shown with a vertical line separating the columns what they're equal to on the right and these are affected when doing RREF or REF
what are elementary row operations?
1. multiply a row by a nonzero constant c
2. interchange two rows
3. add a constant c times one row to another
To return to the original matrix
1. multiply a row by 1/c
2. interchange the same two rows
3. if b = b + cr then b = b - cr
whats is difference between a homogeneous system?
This occurs if a system of linear equations has a constant of all zeros and thus these have the trivial solution
whats is difference between a non-homogenous system?
This occurs if a system of linear equations does not have a constant of all zeros and thus these have the non trivial solution
what is a parametric solution?
using the given vector we can find the x,y,z... depending if the matrix is trivial or non trivial
what is a particular solution?
A(x bar) = (b bar)
what is a general solutoin?
A(x bar) = (zero vector)
what is the definition of a matrix?
tables of numerical data organized into rows and columns
how does one do the addition operation?
add each component of each location
(i.e) (4,5)+(1,4) = (5,9)
how does one do the scalar multiplication operation?
everything is multiplied by that scalar
how does one do multiplication of matrices operation?
think if this -I say A = nxn and B = nxm then AB nxm
how does one do the transpose operation?
to get this the rows and columns must be switched. think of this as if you're rotating the original matrix to the right
what are the properties of matrix operations?
AB != BA
how does one find the inverse of a matrix using gauss jordan elimination?
[A|I] -> [I|A^-1]
how does one find the inverse of a matrix using the determinant?
A^-1 = (1/det)(d, -b, -c, a)
what is the definition of a determinant?
ad = bc
how does one find the determinant using cofactor expansion?
this works like the cross product but i, j, and k are replaced with numbers. Also if this is 4x4 then it will be even larger where you multiply the cross product by the row or column on the outside
how does one find the determinant using the arrow technique?
write out the determinant of the matrix you're working with and add the fist two columns on the right. Go to the top right number in the matrix and draw a downward diagonal to the left and right. then add two downward arrows to the left and right under the original arrows. where the arrows on the right are positive and the arrows on the left are negative. then arithmetic...
how does one find the determinant using elementary row operations?
use RREF where
1. if the matrix is multiplied by a constant, c, the potential determinant is multiplied by c
2. if rows are interchanged the potential determinant is multiplied by -1
3. if a multiple of a row is added the potential determinant is multiplied by 1
what are the properties of determinants?a
if a is nxn with two proportional rows or two proportional columns then det(A) = 0
what is cramers rule?
A(x bar) = (b bar)
(b bar) = (z1,z2)
A1 = (z1 + x, z2 + y)
A2 = (x + z1, y + z2)
x1 = det(A1)
x2 = det(A2)
x = (x1,x2)
what are euclidean vector spaces in R^n ?
when something is a basis and if (vbar)s = (c1, c2, ..., cn) is the coordinate vector of (v bar) relative to S then (v bar) -> (v bar)s creates a connection between vectors is the general vector space and euclidean vector space
[v bar]s = [c1, c2, ...., cn]
describe the addition operation of euclidean vector spaces?
If v and w are vectors in 2-space or 3-space
that are positioned so their initial points coincide, then the two vectors form adjacent
sides of a parallelogram, and the sum v + w is the vector represented by the arrow
from the common initial point of v and w to the opposite vertex of the parallelogram
describe the scalar multiplication of euclidean vector spaces?
a
describe the norm of euclidean vector spaces?
a
describe the dot product of euclidean vector spaces
a
describe the cross product of euclidean vector spaces?
a
whats is orthogonality?
a
what is an orthogonal projection?
a
what is the definition of a linear transformation?
a
what is the domain in relation to a linear transformation?
a
what is the codomain in relation to a linear transformation?
a
what is the standard matrix for a transformation in relation to a linear transformation?
a
what is the matrix transformations in both r2 and r3 for reflection
a
what is the matrix transformations in both r2 and r3 for orthogonal projection?
a
what is the matrix transformations in both r2 and r3 for rotation?
a
what is the matrix transformations in both r2 and r3 for contraction?
a
what is the matrix transformations in both r2 and r3 for dilation?
a
what is composition of transformationsa
...
what is the definition of a vector space?
a
what is a subspace?
a
what does it mean to span?
a
what is linear dependence?
a
what is linear independence?
a
what is basis?
a
what is dimension?
a
what is column space?
a
what is row space?
a
what is null space?
a
what is rank?
a
what is nullity
a
what are the coordinates and change of basis?
a
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