Consider the vectors v1 = [1,1,2],v2 = [1,0,1],v3 =...

Consider the vectors v1 = [1,1,2],v2 = [1,0,1],v3 = [2,1,3]. a) By evaluating an appropriate determinant,decide whether or not the set {v1, v2, v3} is linearly independent. b) Does span {v1, v2, v3} = R3, justify your answer. c) Consider the set W of all real 2 x 2 matrices [a b; c d] with b + c = 1. Is this set a subspace of M2x2? Justify your answer.

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00:05 So we have v1 is 112, v2 is 101, and v3 is 213.

00:14 So these are the three vectors given.

00:19 We need to check whether these are linearly independent.

00:22 So it's enough to find the determinant of this.

00:25 Keep all the vectors in a row form 112, 101, 213.

00:31 So let's find the determinant.

00:34 So we can use the properties of determinants, so 112.

00:37 Subract second row minus first row.

00:39 So it is 0, negative 1 ,000.

00:41 Negative 1, subtract third row, third row is third row minus two times the first row.

00:49 So it's 0 minus 1 minus 1.

00:53 You can see the two rows are identical, so the determinant is 0.

00:58 So that means the vectors are linearly dependent.

01:01 So v1, v2, v3 are linearly dependent, linearly dependent because the determinant is 0.

01:10 Now, the next question is, the span of v1, v2, v3 is r cubed.

01:16 Now i can ignore v3 because v3 is actually depending on v1 and v2.

01:21 And v1 in v2 are obviously linearly independent because two non -parallel vectors are obviously linearly independent.

01:27 Now the question, second question we can frame it as does v1 and v2 no need of v3? does i mean, does span of v1 and v2 can it give rq? let's see.

01:44 What do you mean by span of v1 and v2? lambda into 112 plus some mu into 101...

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