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Linear Algebra Midterm
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Terms in this set (41)
linear system
Consists of 2 or more linear equations in the same variables
homogeneous linear system
All variables equal 0
consistent
Has a solution for every possiblity
augmented matrix
matrix with an additional column
elementary row opperations
swap 2 rows, multiply a row by a constant, multiply one row by a constant and add it to another row
Reduced Row echelon form
each column which has a leading 1 has zeroes everywhere else in the column
If a row has any nonzero entries, then the first nonzero entry is a 1
Any rows of zeroes are at the bottom
In any two rows that are not all zeroes, the leading 1 of the lower row is farther to the right than the leading 1 in the higher row
Row echelon form
If a row has any nonzero entries, then the first nonzero entry is a 1
Any rows of zeroes are at the bottom
In any two rows that are not all zeroes, the leading 1 of the lower row is farther to the right than the leading 1 in the higher row
pivot posistions
Rows with leading 1's
pivot columns
columns that contain leading 1's
leading variable
determined variables, pivots
free variable
columns with no pivots
Gauss-Jordan Elimination
reducing the matrix to reduced row-echelon form
Gaussian Elimination
_____ _____ uses elementary row operations to reduce a linear system of equations to echelon form.
nonsingular matrix
invtertible matrix
inverse matrix
multiplied by the original matrix to get the identity
transposed matrix
rows become columns
equivalent matrices
matrices that can have row operations done on one to get the other
inversion algorithm
multiply (1/ad-bc) by a matrix that is
d -b
-c a
diagonal matrix
every off diagonal entry is zero
triangular matrix
all entries above or below the diagonal are zero
symmetric matrix
aij = aji for every i and j
cofactor expansion
aka Expansion by Minors. Pick a row or column and multiply each entry in that row/column by the determinant of what is left
How ERO's affect determinant
row swap - take negative
addition - no change
multiply by n - determinant/n
THM 2.3.8
15 Equivalent Statements
10 axioms of a vector space
check closure
find 0 vector element
check the rest
subspace
a subset that is also a vector space
THM 4.2.1
If w is a nonempty subset of a vector space v, then w is a subspace of v iff
when u,v are in w the (u+v) are in w
if k is a scaler and u is in w then ku is in w
linear combination
Let w, v1, v2.... vn be vectors in a vector space V, and k1, k2...kn be scalers. W can be expressed as k1
v1 + k2
v2...kn*vn.
span
All of the linear combinations of a set of vectors
linearly independent
vectors that cannot be made as linear combinations of the others
basis
a set of linearly independent vectors that spans a space
coordinate vector
constants of linear combination written as a vector
THM 4.4.1
Every vector in V can be expressed uniquely as a linear combination of basis vectors
dimension of a space
a finite dimensional vector space with n elements in it's basis has a ______ of n
transition matrix
Change of basis matrix
row space
The subspace spanned by the row vectors of a matrix
column space
subspace spanned by the column vectors of a matrix
null space
the solution space to Ax = 0
rank
Number of leading 1's in rref(A). Equal to the dimension of the row space and the dimension of the column space
nultility
Number of free variables in rref(A)
4 Fundamental Spaces of a matrix
row(A), col(A), null(A), and null(A^T)
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