4. I. If S is linearly independent and T is a subset of S, then T is linearly in

Question

4. I. If S is linearly independent and T is a subset of S, then T is linearly independent. II. If S is linearly dependent and T is a subset of S, then T is linearly dependent. a) I only b) Il only c) I and Il d) neither I nor lI. 5.1. Ix2 is a solution to an rh order linear homogeneous differential equation with constant coefficients, then so is 1. Il. . If 1 is a solution to an nth order linear homogeneous differential equation with constant coefficients, then so is x. a) I only b) ll only c) I and II d) neither I nor I1.

Explanation / Answer

4. Option A is correct ,i.e., only statement 1 is correct

Statement 1: Suppose we have a finite linearly independent set S of vectors, then, by definition, for all choices of distinct elements v1,v2,?,vk in S (i.e. every subset of distinct elements) and scalars c1,c2,?,ck,
?ki=1civi=0 implies c1=c2=?=ck=0
That means that no vector in these subsets could ever be a linear combination of other two or more vectors in that same subset of S, therefore they are linearly independent, as is the subset itself.

Statement 2: take the set of vectors b1=(0,0,1), b2=(0,1,0) and b3=(0,1,1) as a basis. This is a linearly dependent set since you can get b3 as a linear combination of b2 and b1. However, the subset {b1,b2} is linearly independent, since you cannot get b1 as a linear combination of b2, or vice-versa

Related Questions

Navigate

• Browse (All)
• Browse 4
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.