1. Assuming the standard definitions of addition and scalar...

1. Assuming the standard definitions of addition and scalar multiplication, decide whether or not the following sets are vector spaces. Justify your answers by verifying that all properties of a vector space are satisfied or that at least one of the properties fails to hold. (a) The set V of all 3 × 3 lower triangular matrices (all entries above the main diagonal are zero). (b) The set V of all 2 × 2 matrices whose determinants are 0.

Transcript

00:01 So for this question, we have two sets of matrices, basically, and we want to determine whether or not they're vector spaces.

00:07 So we want to assume the standard definitions of addition and scaling multiplication.

00:11 And again, just decide whether or not these sets are vector spaces.

00:14 And we should need to justify the answers.

00:17 So let's see.

00:19 In part a, our set here, the set v, this is supposed to be three by three matrices that are upper, or sorry, that are lower terrain.

00:33 So matrices with all entries above the diagonal are zero.

00:47 So what that looks like basically, it's a 3x3 matrix where i can have anything i want on the diagonal.

00:54 We can have anything we want below the diagonal, that's the lower triangular bit, but we better have zeros above the diagonal.

01:02 So they're just matrices that have this form.

01:05 And we want to know under the standard operations, does this same? set give you a vector space.

01:11 So first of all, it contains zero, right? because, you know, the stars, they're allowed to also be zero.

01:19 They just don't have to be zero.

01:21 Where i wrote the zeros, those entries are only allowed to be zero.

01:25 So what that means for us is that we do contain the zero.

01:29 So zero is in my set v.

01:32 Also the identity matrix, right, where i have ones on the diagonal.

01:37 And zeros elsewhere, the identity is in this space, so that's good.

01:45 Those are two requirements for vector space.

01:48 Then if you take a scale of multiple, so if i took like some alpha is a real number, and i multiply it by a matrix of this form, well, you still get something, you know, maybe you get different values, you get alpha times whatever the star values are, but alpha times zero is zero, right? so you still have this condition that those upper entries are zero.

02:14 Scalar multiplication doesn't change that.

02:16 So if i take a scalar multiple of one of these elements, i land back inside the set, so that's good.

02:25 Again, if i add two of these things, right, if you add matrices that have this form, in those upper, you add matrices entry -wise.

02:34 So those zeros being in the upper entries, you're just getting zero plus zero for the entry.

02:40 So adding two of these elements lands you back in the set as well.

02:44 So the sets closed under basically v is closed under scale or multiplication and addition, those two symbols...

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