Vocab Flashcards
linear equation
an equation that can be written in the form a1x1 + … + anxn = b with b and the coefficients a1…an are real/complex numbers
system of linear equations/ linear system
collection of one or more linear equations involving the same variables
linear systems are equivalent if
they have the same solution set
two matrices are row equivalent
if there is a sequence of elementary row operations that transforms one matrix to another
leading entry
the leftmost nonzero number in a nonzero row
basic variables
variables that correspond to pivot columns
vector / column vector
matrix with only one column
two vectors are equal if
their corresponding entries are equal
linear combination
y = c1v1 + … + cnvn given vectors v1,…,vn, and scalars c1,…,cn
homogeneous
Ax=0
trivial solution
- when x=0
- every homogeneous equation has the trivial solution
nontrivial solution
- x is nonzero
- occurs when there is a free variable
linearly independent
set of vectors {v1, … , vp} with the vector equation x1v1+ … + xpvp = 0 with only the trivial solution
linearly dependent
there exists weights c1, …, cp not all zeros such that c1v1 + …+ cpvp = 0
transformation / function / mapping
a rule that assigns each vector x in Rn a vector T(x) in Rm
Rn
domain
Rm
codomain
T(x)
image
range
set of all T(x)
a transformation is linear if
T(0) = 0 T(u+v) = T(u) + T(v) T(cu) = cT(u)
onto
each b in Rm is the image of at least one x in Rn
one-to-one
each b in Rm is the image of at most one x in Rn
two matrices are equal if
they have the same size (same number of rows and columns)
nonsingular matrix
invertible matrix