Let v_1, v_2, v_3 be the vectors in R^3 defined by v_1 = [-28 29 -2] v_2 = [12 -
ID: 3036370 • Letter: L
Question
Let v_1, v_2, v_3 be the vectors in R^3 defined by v_1 = [-28 29 -2] v_2 = [12 -11 13] v_3 [16 -18 -11] (a) Is {v_1, v_2, v_3} linearly independent? Write all zeros if it is or if it is linearly dependent write zero as a non-trivial (not all zero coefficients) linear combination of v_1, v_2, and v_3 (b) Is {v_1, v_2} linearly independent? Write all zeros if it is or if it is linearly dependent write zero as a non-trivial (not all zero coefficients) linear combination of v_1 and v_2 (c) Type the dimension of span {v_1, v_2, v_3}:Explanation / Answer
Let A =
-28
12
16
29
-11
-18
-2
13
-11
We will reduce A to its RREF as under:
Multiply the 1st row by -1/28
Add -29 times the 1st row to the 2nd row
Add 2 times the 1st row to the 3rd row
Multiply the 2nd row by 7/10
Add -85/7 times the 2nd row to the 3rd row
Add 3/7 times the 2nd row to the 1st row
Then the RREF of A is
1
0
-1
0
1
-1
0
0
0
(a) Apparently, the set {v1,v2,v3} is linearly dependent and v3 = -v1-v2 or, v1+v2+v3 = 0
(b) The set {v1,v2} is linearly independent.
(c) The dimension of the span {v1,v2,v3} is 2 as the set {v1,v2,v3} is linearly dependent and the set {v1,v2} is linearly independent.
-28
12
16
29
-11
-18
-2
13
-11
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.