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Terms in this set (14)
Image of a linear transformation
The span of the vectors of the linear transformation
ex: [1,0;0,2;0,1][s;t] = [s;2t;t] = Im(A)
Orthogonal Complement of a Subspace
Le U = subspace of R^n. Then, the orthogonal component of U is the set U^ortho = {v in R^n : v*u = 0 for all u in U }
Orthogonal vectors
vectors are orthogonal if the dot product between them is zero. there's a symbol for this it looks like perpendicular symbol because it is
Basis of a Vector space
A basis of a vector space V is a subset {v1, ... vl} of V s.t
1) Span {v1, ..., vl} = V
2) {v1,...,vl} is linearly independent
Subspace W of a vector space V
A subset W of V is a subspace of V if
1) 0 is in W
2) W is closed under vector addition
3) W is closed under scalar multiplication
Null Space of a matrix A
All {x : Ax = 0}
Column space of a matrix A
the span of the linearly independent columns of A
Invertible Matrix
An nxn matrix is called invertible iff there exists an nxn matrix C s.t AC = CA = In
where In is the nxn Identity Matrix
Linear Transformation T: V->W
A transformation T is linear iff
1) for all x,y in V, T(x + y) = T(x) + T(y)
2) for all x in V, T(ax) = aT(x)
Linearly independent/dependent vectors
A set of vectors {v1, ...., vn} is said to be linearly independent iff the equation
c1v1 + ... + cnvn = 0 has only the trivial solution.
if there is a nontrivial solution, the vectors are linearly dependent
Eigenvalue of an nxn matrix A
A scalar λ is said to be an eigenvalue of A if there is a nontrivial x s.t
Ax = λx
Eigenvector of an nxn matrix A
A vector x is said to be an eigenvector of A if there is a nontrivial λ s.t.
Ax = λx
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