Consider the following 4 vectors in R^3: a = [1 -1 2], b = [-1 0 3], v = [-1 -2
ID: 3034961 • Letter: C
Question
Consider the following 4 vectors in R^3: a = [1 -1 2], b = [-1 0 3], v = [-1 -2 13], w = [0 -1 1]. Is v in the span on a and b? Is w in the span on a and b? Clearly justify your answers. Problem 2. Find the value of h for which the following vectors are linearly dependent: [3 -6 1], [-6 4 3], [9 h 3]. Problem 3. Consider the following subsets of R^3: V_1 = {[a b c] with a + b + c = 1}, V_2 = {[x_1 x_2 x_3] with x_1 - 2x_2 + x_3 = 0}, V_3 = {[a a + 1 b] with a, b any real numbers} For each of them, decide whether they are subspaces of R^3. Clearly justify your answers.Explanation / Answer
3) V1 { a+b +c = 1 )
closed under addition : a1 +b1 +c1 -1 =0 ; a2 + b2 + c2 -1 =0
add the two : a1+a2 + b1+b2 + c1+c2 - 2 =0 ; ( a1 + b1 +c1 -1) + a2 +b2 +c2 -1) =0
closed under scalar multiplication : k(a1) + k(b1) + kc1 -1 =0 ; k( a1+b1 +c1 - 1/k) =0
This is not equal to k(a1 +b1 +c1 -1) =0
Not closed undersscalar multiplication
No zero vector in subset
Not a subspace
V2 { x1 -2x2 + x3 =0}
closed under addition , cliosed under scalar multiplication , zero vector is in the subset
Hence is a subspace
V3 { a, a+1 , b}
closed under addition
zero vector is not in the subspace : { 0, 1, 0}
Not a subspace
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