Linear Algebra Midterm 2

Let {____} be a set of vectors. We say that {____} is linearly independent if, for all ______ E R, if _______ = 0, then c1,…,cn = _.

v1, v2, …, vn; v1, v2,…vn; c1, c2,…, cn; c1v1 + c2v2 +…+ cnvn; 0

Let {____} be a set of vectors. We say that {____} is linearly dependent if, there are scalars, c1…,cn E R not all zero such that _____ = 0.

v1, v2,…, vn; v1, v2,…,vn; c1v1 + …+cnvn

Linear Combination

A vector w is a linear combination of the vectors v1,.., vn in a vector space V if there exist scalars c1,…,cn E R such that w = c1v1+…+cnvn.

Span

If v1,…, vn are vectors in a vector space V, then the span of v1,…,vn is the collection of linear combination of v1,…, vn.

Basis

The set of vectors {v1,…,vn} is a basis for a vector space V if it is linearly independent and its span is V.

Dimension

If V is a vector space over F, the dimension of V is the number of vectors in any basis of V.

Row Space (A is a matrix)

The row space of A is the subspace of F1xn spanned by the rows of A.

Column Space (A is a matrix)

The column space of A is the subspace of Fmx1 spanned by the columns of A.

Nullspace (A is a matrix)

The nullspace of A is the set of all x E Fmx1 such that Ax = 0.

Nullity

The nullity of A is the dimension of the nullspace of A.

Rank

The rank of A is the dimension of the row space or column space of A.

Linear Transformation

If V and W are vector space over F, then a linear transformation is a function T: V→W such that, for all c E F and v, w E V, we have T(cv + w) = cT(v) + T(w).

Injective

A function f: A→B is injective if, for all a, a’EA, if f(a) = f(a’), then a = a’.

Surjective

A function f: A→B is surjective if, for all b EB, there is a E A such that f(a) = b.

Bijective

A function f:A→ B is bijective if it is both surjective and injective.

Isomorphism

If V and W are vector spaces over F, then a function T: V→ W is an isomorphism if it is a bijective linear transformation.

Kernel

If T: V→W is a linear transformation, then the kernel of T is the set of all vEV such that T(v) = 0.

Image

If T:V→ W is a linear transformation, then the image of T is the set of all wEW such that there exists vEV for which T(v) = w.

Coordinate of V with respect to B

Let V be a finite-dimensional vector space over F with given basis B = {B1,…,Bn}. If v E V then the coordinate of v with respect to B is given by the vector (c1, c2,…, cn)^t E Fnx1 such that v = c1B1 +…+ cnBn.