Question

Let the vectors a and b be in X = Span{x1,x2,x3}. Assume all vectors are in...

Let the vectors a and b be in X = Span{x1,x2,x3}. Assume all vectors are in R^n for some positive integer n.

1. Show that 2a - b is in X.

Let x4 be a vector in Rn that is not contained in X.

2. Show b is a linear combination of x1,x2,x3,x4.

Edit: I don't really know what you mean, "what does the question repersent." This is word for word a homework problem I have for linear algebra.

Homework Answers

Answer #1

1. Since the vectors a and b are in Span {x1,x2,x3}, hence both a and b are linear combinations of x1,x2,x3. Let us assume that a = a1x1+a2x2+a3x3 and b = b1x1+b2x2+b3x3 where the ais and bjs are real valued scalars. Then 2a-b = 2(a1x1+a2x2+a3x3) –(b1x1+b2x2+b3x3) = (2a1-b1)x1 +(2a2-b2)x2 +(2a3-b3)x3. Thus, 2a-b, being a linear combinations of x1,x2,x3 , is in X = Span{x1,x2,x3}.

2. Let x4 be a vector in Rn that is not contained in X. Then x4 is not a linear combination of x1,x2,x3. Now, b = b1x1+b2x2+b3x3 = b1x1+b2x2+b3x3+0x4 . This shows that b is a linear combination of x1,x2,x3,x4.

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