Two Matrix Questions concerning Col and Nul Space

http://i.imgur.com/CLTa4FX.png

I'm not entirely sure how to go about answering these questions. for 6a, I think it can be explained through a theorem, something about the number of rows and variables

But for the rest, I'm not actually sure what is being asked. This section of my course has posed the most difficulty for me, because I have no access to lectures (extramural student), and there was literally only a page of explanation followed by about 40 questions. I've been through all the Khan Academy stuff, but I don't really get what is being asked here. Any tips on how to proceed? I can do all the row reduction and basic row operations, but how should I go about using them?

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Re: Two Matrix Questions concerning Col and Nul Space

I'm not sure whether you understand clearly what linear independence is about. I assume you know the formal definition. The "basic idea" is that none of these three (or perhaps more, in another case) vectors can be constructed by adding together any multiples of the other two (or perhaps more). If you think of these vectors as arrows or directed line segments from the origin, it means that there is no plane that contains all three. Two vectors in a plane can be used to construct another vector in the same plane, but not any vector that isn't in that plane. Is that clear?

In this case, B contains two vectors of three elements. The only way two vectors (of however many elements) can be linearly dependent is if each is a multiple of the other. Is that the case here? No. You should be able to see that 1:2 is not the same as 4:7.

Now if vector u can be made by combining v1 and v2, it is in the span of B. The way to find out is to construct a 3x3 matrix where the columns are v1, v2 and u and row-reduce them. If they come out with a row of zeroes at the bottom, then u is in the span of B.

See the pdf.

Let me know if I lost you anywhere.

If you got this, then we can get to work on the other questions.

Re: Two Matrix Questions concerning Col and Nul Space

Yep, but the first one isn't really my problem. The other 3 parts are 6b, 6c and 7

Re: Two Matrix Questions concerning Col and Nul Space

OK, the fact that we get a row of zeroes at the bottom is enough to show that u belongs to that span. I assume you understand why. If you don't, let me know, I'll try to explain.

Now as for the coordinate vector, what you need is two scalars, we can call them x and y, and we'll arrange them in a column vector like this

x

y

... we need to choose them so that x times v1 + y times v2 = u. You need a two-entry column vector. Do you see why it's two entries, not just one, and not three or more? If you do, and you can see the pdf I gave you, can you find the values of x and y?

Re: Two Matrix Questions concerning Col and Nul Space

I understand why B is linearly independent and have the answer to 6a

I can do the row reduction like in your PDF (I worked it out myself) and came to the same numbers you had. But I don't actually understand what 6b is asking. What does it mean by u exists in H = span (B)?

Re: Two Matrix Questions concerning Col and Nul Space

The span of B is the set of all vectors that can be expressed as linear combinations of the vectors in B. That is, u is in the span if there are numbers x and y such that x times v1 plus y times v2 equals u. Do such numbers exist? If so, what are they? If you've row reduced the 3x3 augmented matrix, you should see the answer in front of you.

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Re: Two Matrix Questions concerning Col and Nul Space

I have to go to sleep (middle of night here), but check the new pdf. I hope I explained it well enough. The rank of A is 3 and the nullity is 0, unless a = 1, in which case rank(A)=2 and nul(A)=1.

Oops, I was just about to go to sleep when I had a thought. I calculated the determinant, and came up with a^2 - 1 = 0. What if a = -1? See the pdf. Appears in that case we'd have a rank of one and a nullity of two. I must've goofed on my row-reduction, but I'm too tired to track down the error. Maybe you can do that, let me know if you find it.

Re: Two Matrix Questions concerning Col and Nul Space

Quote:

Originally Posted by

**zhandele** The span of B is the set of all vectors that can be expressed as linear combinations of the vectors in B. That is, u is in the span if there are numbers x and y such that x times v1 plus y times v2 equals u. Do such numbers exist? If so, what are they? If you've row reduced the 3x3 augmented matrix, you should see the answer in front of you.

Alright, I get this. I just hadn't actually seen this notation, so it is actually saying u exists in "H which has the basis Span B". I was confused by the equals sign.

Re: Two Matrix Questions concerning Col and Nul Space

Looking over your Row Reduction, how did you get the final jump into RREF? I can't see how you got rid of the a+1 and 1-a on row 2 (for question 7)