Proof of vector dimensions using inequalities

[TheFallen018]

Hello all!

I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is.

It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove it? I can't think of a good way to do that.

The reason for my thinking that there are only 2 linearly independent vectors in the span, is that if there were less, w1 and w2 would have to be multiples of each other. If they were multiples of each other, it would be impossible for v1 and v2 to be independent. Also, there can't be more than 2 dimensions, because there are 3 vectors in the span, but only two are independent. I'm not convinced that this explanation would do the trick for the problem though. Is there a better way to show this thinking?

Any ideas would be great!

Thanks

 

[Steven G]

Hello all!

I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is.

It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove it? I can't think of a good way to do that.

The reason for my thinking that there are only 2 linearly independent vectors in the span, is that if there were less, w1 and w2 would have to be multiples of each other. If they were multiples of each other, it would be impossible for v1 and v2 to be independent. Also, there can't be more than 2 dimensions, because there are 3 vectors in the span, but only two are independent. I'm not convinced that this explanation would do the trick for the problem though. Is there a better way to show this thinking?

Any ideas would be great!

Thanks

What are you referring to? Can we have the entire problem? That would be very helpful.

 

[TheFallen018]

What are you referring to? Can we have the entire problem? That would be very helpful.

Sure, here it is. Thanks

 

[Steven G]

Sure, here it is. Thanks

View attachment 10713

We know that w1, w2 and w3 span W. We are now given that w3 is a linear combination of w1 and w2. Hence w1 and w2 span W. This tells us that Dim W <=2 (after all, maybe w2 is a linear combination of w1). We also know that v1 and v2 are linear independent--but may NOT span W. So Dim W >=2. Combining yields that Dim W = 2