Linear Algebra

Consider vectors v1,...,vp and w1,...,wq in a subspace V of Rn. If the vec- tors v1,...,vp are linearly independent, and the vectors w⃗1,...,w⃗q span V, then q ≥ p.

consider vectors...

All bases of a subspace V of Rn consist of the same number of vectors.

all bases...

Independent vectors and spanning vectors in a subspace of Rn
Consider a subspace V of Rn with dim(V ) = m.
a. We can find at most m linearly independent vectors in V .
b. We need at least m vectors to span V .
c. If m vectors in V are linearly independent, then they form a basis of V . d. If m vectors in V span V, then they form a basis of V.

independant vectors...

To construct a basis of the image of A, pick the column vectors of A that correspond to the columns of rref(A) containing the leading 1’s.

to construct...

for any matrix a dim(im A) = rank (A)

for any matrix a ..

For any n × m matrix A, the equation
dim(ker A) + dim(im A) = m
holds. The dimension of ker(A) is called the nullity of A, and in Theorem 3.3.6 we observed that dim(imA) = rank(A). Thus, we can write the preceding equation alternatively as
(nullity of A) + (rank of A) = m.
Some authors go so far as to call this the fundamental theorem of linear algebra.

for any n x m

Suppose you are able to spot the redundant columns of a matrix A.
Express each redundant column as a linear combination of the preceding columns, v⃗i = c1v⃗1 + ··· + ci−1v⃗i−1; write a corresponding relation,
−c1v⃗1 −···−ci−1v⃗i−1 +v⃗i =0⃗;and generate the vector in the kernel of A. The vectors so constructed form a basis of the kernel of A. The nonredundant columns form a basis of the image of A.
The use of Kyle numbers can facilitate this procedure. See Example 3.

suppose you are able

Bases of Rn
The vectors v⃗1,...,v⃗n in Rn form a basis of Rn if (and only if) the matrix
⎡| |⎤ ⎣v⃗1 ··· v⃗n⎦ ||
is invertible.

bases of Rn

Linearity of Coordinates
If􏰰is a basis of a subspace V of Rn ,then 􏰇􏰈􏰇􏰈􏰇􏰈
a. x⃗+y⃗􏰰= x⃗􏰰+ y⃗􏰰, forallvectorsx⃗andy⃗inV,and 􏰇􏰈􏰇􏰈
b. kx⃗ 􏰰 =k x⃗ 􏰰, forallx⃗inV andforallscalarsk.

Linearity of coordinates