x^2-1=0
Γ(s)
EXAMPLE 3 Determine if the following sets of vectors are linearly independent.
a. (v_1)=[[3],[1]],(v_2)=[[6],[2]]
b. (v_1)=[[3],[2]],(v_2)=[[6],[2]]
(v_1)+(v_2)=
sssss
SOLUTION
a. Notice that (v_2) is a multiple of (v_1), namely (v_2)=2*(v_1). Hence −2*(v_1)+(v_2)=0, which shows that {(v_1),(v_2)} is linearly dependent.
b. The vectors (v_1) and (v_2) are certainly not multiples of one another. Could they be linearly dependent? Suppose c and d satisfy
c*(v_1)+d*(v_2)=0
If c≠0, then we can solve for (v_1) in terms of (v_2), namely (v_1)=(−d/c)*(v_2). This result is impossible because (v_1) is not a multiple of (v_2). So c must be zero. Similarly, d must also be zero. Thus {(v_1),(v_2)} is a linearly independent set.
The arguments in Example 3 show that you can always decide by inspection when a set of two vectors is linearly dependent. Row operations are unnecessary. Simply check whether at least one of the vectors is a scalar times the other. (The test applies only to sets of two vectors.)
A set of two vectors {(v_1),(v_2)} is linearly dependent if at least one of the vectors is a multiple of the other. The set is linearly independent if and only if neither of the vectors is a multiple of the other.
a=0
b=1
a+b=1
sin(x)′=cos(x)