Spans, Linear Independence, and redundant
vectors – Theory notes for quiz prep
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Set of all linear combinations of vectors u1,u2…uk in Rn is known as the span of these
vectors.
If K=0, the span only consists of the zero vector also called empty linear combination
{0}
If span {u,v,w} = span {u,v} then w is redundant
1 or more redundant vector = linearly dependent
No redundant vector = linearly independent
U1….uk are linearly independent if and only if homogeneous equation a1u1 + …. +
akuk = 0 has only the trivial solution
Linear dependence and reordering: if vectors are linearly dependent, then so is any
reordering sequence.
Linear independence of a subspace: if u1, u2 … uk are linearly independent then so are
u1, u2 … uk for any value of j>k
Linear independence and dimension: if u1,u2, … uk is a sequence of vectors in Rn, then
vectors are linearly dependent for k>n.
If A is invertible, then the set of vectors made of the columns of A is linearly
dependent.
U1,u2,… uk are linearly independent. Then every vector v spans{u1,u2,…uk} can be
written as a linear combination of u1, u2, …. Uk in a unique way.
Span {uji…ujl} = span {u1 … uk} they are linearly dependent.
where the first term is a subsequence of vectors by removing all redundant vectors
and the second is a sequence of vectors and they two terms are linearly dependent.
V is a subspace of Rn if:
1. V contains zero vector
2. V is closed under addition
3. V is closed under scalar multiplication
V = { x ϵ Rn | Ax = 0} this is the solution space of the system. Here, V is the set of
solutions is a subspace of Rn and Ax=0 is a system of homogenous equations.