Final Exam Flashcards
1.1 A system of linear equations has how many soluntions
no solution, exactly one solution, or infinite solutions
1.1 consistent
system has a solution
1.1 inconsistent
system has no solution
1.2 Free variable
no pivot in this row x3 = x3
1.2 Existence and Uniqueness Theorem: A system is consistent only if
the augmented column is not a pivot column
1.2 If a linear system is consistent then
it has a unique solution when there’s no free variables, it has infinite solutions when there is at least 1 free variable
1.3 Parallelogram Rule
0, u, and v are points on a plane. Then the 4th point is u+v
1.3Linear combination
y = c1v1 + … + cpvp
1.3 Span
set of all linear combinations of v1….vp
1.4 Theorem 3: The equation Ax=b has the same solution set as ____
x1a1 + ……xpap
Ax = b has a solution( is consistent) only if ____
b is a linear combination of the columns of A.
1.5 Homogeneous
system can be written in the form Ax =0
1.5 Trivial Solution
solution where x =0
1.5 Nontrivial solution
a nonzero vector x that makes Ax =0
1.6 Linearly Independent
a set of vectors is linearly independent if x1v1+…xpvp =0
1.7 Linearly Dependent
A set of vectors is linearly dependent if there are nonzero constants that make c1vi+…cpvp=0
1.7 Theorem: The columns of A are linearly independent if what
The equation Ax=0 has only the trivial solution (where x=0)
1.7 A set of 2 vectors is linearly dependent if what
if at least one of the vectors is a multiple of the other.
1.7 A set is linearly independent if what?
if the vectors are not multiples of one another
1.7 Characterization of Linearly Dependent sets: An set of 2+ vectors is linearly dependent if
if one of the vectors is a linear combination(y=c1v1) of the others or if v1=0.
1.7 If a set contains more vectors than there are entries in each vector, then…
the set is linearly independent
1.8 domain
The set R^n
1.8 codomain
the set R^m
1.8 image
the vector T(x) in R^m
1.8 range
the set of all images T(x)
1.8 if T is a linear combination then T(cu+dv)=?
cT(u)+dT(v)
1.8 A transformation T is linear if: (hint 2)
T(u+v) = T(u)+T(v)
T(cu) = cT(u)
1.9 Let T: R^n-> R^m be a linear transformation. Then there exists a unique matrix A such that
T(x) = Ax
1.9 Onto
pivot in every row
1.9 one to one
pivot in every column
1.9 A linear transformation is one to one only if
T(x) = 0 has the trivial solution (x=0)
1.9 T maps R^n onto R^m only if
the columns of A span R^m
1.9 T is one to one only if
the columns of A are linearly independent
2.1 (matrix multiplication) If A is an mxn matrix and B is an nxp matrix, the the product of AP is what
the mxp matrix whose columns are [Ab1…..Abp]
2.1 Each column of AB is what
a linear combination of the columns of A using the constants from the corresponding column of B
2.1 (A^T)^T =?
A
2.1 (A+B)^T = ?
A^T + B^T
