Theorem: if A is a m x n matrix with cols a1,....,an, and if x is in Rn,
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Then Ax is the linear combination of the cols of A using the corresponding
entries of x as weights.
Ax = [a1 a2 ... an][x1 ... xn] = x1a1 +x2a2 +... +xnan
Ax = b = x1a1 + ... + xnan = [ a 1 ... an b]
, A(u + v) = Au + Av
A(cu) = cA(u)
When does a homogenous system have a nontrivial solution?
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When there is a free variable in Ax
Linear independence
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An indexed set {v1...vp} in R^n is said to be linearly independent if the
vector equation c1v1 +...+cpvp= 0 has only the trivial solution.
Linear Combination
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A sum of scalar multiples of vectors. The scalars are called the weights.
c1v1+c2v2+...+ckvk
Thm. 9 if S contains the 0 vectors
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Give this one a try later!
Then Ax is the linear combination of the cols of A using the corresponding
entries of x as weights.
Ax = [a1 a2 ... an][x1 ... xn] = x1a1 +x2a2 +... +xnan
Ax = b = x1a1 + ... + xnan = [ a 1 ... an b]
, A(u + v) = Au + Av
A(cu) = cA(u)
When does a homogenous system have a nontrivial solution?
Give this one a try later!
When there is a free variable in Ax
Linear independence
Give this one a try later!
An indexed set {v1...vp} in R^n is said to be linearly independent if the
vector equation c1v1 +...+cpvp= 0 has only the trivial solution.
Linear Combination
Give this one a try later!
A sum of scalar multiples of vectors. The scalars are called the weights.
c1v1+c2v2+...+ckvk
Thm. 9 if S contains the 0 vectors
Give this one a try later!
