4.t. utk "
Let V be an arbitrary nonempty set of obj ects on which hvo operations are defined: I
addition, and multiplication by scalars (numbers). By addition we mean a rule for I
associating with each pair of objects u and v in
A vector w is called a linear combination of the vectors v1,V2, . , Vr if it can be
expressed in the form
where kl' k2, . , kr are scalars.
If r = 1, then the equation in the preceding definition reduces to w = k1V l ;
I I I .
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Functions from R n
Recall that a functi on is a rule I that associates with each element in a set A one and
only one element in a set B. If I associates the element b with the element a, then we
write b = I (a) and say that b is the image of a unde
.AI! bases.for a finite-dim'ensionat vector space have the same number ofvectors.
To see how this theorem is related to the concept of "dimension,'" recall that the stan
dard basis for RIl has n vectors (Example 2). Thus Theo
Elimination Methods We have just seen how easy it is to solve a system oflinear equations once its" augmented
matrix is in reduced row-echelon form. Now we shall give a step-~Umiii1triUIl
procedure that can be useq to reduce any matrix to reduced
The Steps in Solving Selected-Order Euler Equations
Here are the basic steps for ﬁnding a general eelutien to an}:r second-order Euler equation
axEy" + ﬁxy' + y}: = I] for xevﬂ
Remember a , ﬂ anti 7/ are teal-valued constants. Te illustratethe basic metho
Since this result is virtually self-evident when one looks at numerical examples, we shall
omit the proof; the proofinvolves little more than an ~ysis of the positions of the O's
and l's of R.
EXAMPLE 5 Bases f9r Rowand Column Spaces
Second Order Differential Equations
In the previous chapter we looked at ﬁrst order differential equations. In this chapter we will
move on to second order differential equations. Just as we did in the last chapter we