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Linear Algebra 4.1-2


Eight rules for a linear space or vector space associative, commutative, neutral element,has negative, distributive, associative with products, same as mulitple of 1
A linear space is a set with two reasonably defined operations, addition and scalar multiplication, that allow us to form linear combinations.
The neutral element for an nxm matrix the 0 matrix
The neutral elemtn for the linear equations in three unknowns: ax+by+cz=d is a=b=c=d=0
The neutral element of a vector the 0 vector
K(a+ib)=ka+i(kb) form a linear space and have the neutral element 0=0+0i
Subspaces: A subset W of a linear space V is called a subspace of V if a. contains the neutral element 0 of V b. is closed under addition c. W is closed under scalar multiplication (b/c is closed under linear combinations)
Differentiable functions form a subspace, True or False? True
Examples of subspaces c^(infinity), P the set of all polynomials, Pn the set of all polynomials of decree less than or equal to n
we say that f1....f2 span V if every f in V can be expressed as a linear combination of f1...f2
We say that fi is redundant if it is a linear combinationf f1...fi-1. The elements are called linearly independent if none of them is redundant.
If equation c1f1+...+cnfn=0 has only the trivial solution c1=...=cn=0
We say that f1...fn are a basis of V if they span V and are linearly independent. This means that every f in V can be written uniquely as a linear combination.
The coefficients c1...cn are called the coordinates of f with respect to the basis B=(f1...fn)
B-coordinate transformation is invertible 1 True
[f+g]B=[g]B+[f]B T/F T
Finding a basis of a linear space a. Write down a typical element of V, int terms of some arbitrary constants b. Use arbitrary constants as coeff, express typical element as a linear combination of some elements of V c. Verify the elements are linearly independent
Finite dimensional linear spaces A linear space V is called finite dimensional if it has a (finite)basis so that we can define its demenension dim(V)=n
The image is a subspace of the and the kernel is a subspace of the target-domain
If the image of T is finite dimensional, then dim(imT) is called the rank of T
If the kernel of T is finite dimensional, then dim(kerT) is called nullity of T
dim(V)= rank(T) +nullity(T)
dim(V)= dim(imT)+dim(kerT)
Show linear transformation show that closed under scalar multiplication and addition
The kernel consists of all inputs into T that produce 0 as output
Is T 1-1 check Ker(T) must =0
2out of 3 Rule -dim(V)=dim(W) -ker(T)=0 (one-to-one) -im(T)=W (onto)
If one of the three is false then Not isomorphic
If 2 out of three is true Isomporphic
P2-> LB and LC-> R^3
Sc->B = (SB->C)^-1
Isomorphic same structure
Created by: ok2bpure