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I'm proceeding through MIT OCW's 18.06 class in Linear Algebra, and I've reached a sticking point on the first lecture - I was wondering if someone can offer some clarification on a specific point for me.

My confusion is on the explanation behind 'how' you get away with creating a column vector out of the coefficients of the same variable in a system of equations, and then use those same coefficients to become 'motion' along a different coordinate axes.

For instance, in the system of equations below,

\begin{matrix} 1x && + && 2y && = && 2 \\ -3x && + && 4y && = && 5 \\ \end{matrix}

Professor Strang comes up with three separate column vectors. For x, it is: \begin{bmatrix} 1 \\ -3 \\ \end{bmatrix} and for y, it is:

\begin{bmatrix} 2 \\ 4 \\ \end{bmatrix} Similarly, he derives the (2,5) vector as the answer vector which we then take linear combinations of the previous two to determine a solution - I fully grasp all the mechanics of how this works, but DO NOT understand how he can use the 'x' vector (with components 1 & -3) to draw a vector of '1' unit in the x direction and '-3' units in the y direction using nothing but coefficients that came from the x variable! This is the crux of my confusion.

I would have an easier time understanding what was going on if the actual mechanics of creating the two vectors was to take the coefficients of each equation and put those into column vectors. For instance, if it were as follows:

X = \begin{bmatrix}1 \\ 2 \\ \end{bmatrix} and Y = \begin{bmatrix}-3 \\ 4\\ \end{bmatrix}

IF it were this way, it would make sense to me because it would correspond to the 'movement' each equation produced in each coordinate system - I also realize that this 'breaks' the idea of 'x' and 'y' because in this second example (in which I realize my understanding is wrong) I've arbitrarily assigned the 'labels' x and y to the separate vectors.

Can someone offer an explanation?

Thanks,

Luke

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    The lowercase $x$ and $y$ here play the roles of coordinates with respect to the basis $$ {\bf b}_{1} = [1, \; -3]^{T}. \; {\bf b}_{2} = [2, \; 4]^{T}. $$ So, the problem is to find such coefficients $x, y$ that the linear combination $x{\bf b}_{1} + y {\bf b}_{2}$ equals $[2, \; 5]^{T}$. – 2017-01-31
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    Thanks avs, Some serious lights are clicking on for me right now - much appreciated. – 2017-01-31
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    I think you are misunderstanding when you say "for x it is $\begin{bmatrix}1 \\ 2 \end{bmatrix}$", "for y it is $\begin{bmatrix}-3 \\ 4 \end{bmatrix}$". Those are NOT values for x and y. x and y are *numbers* not vectors. Those column vectors are the coefficients of x and y, not the values of x and y themselves. – 2018-04-04

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Without having worked through the course myself, this appears to be more a product of confusing notation than of a deeply rooted misunderstanding.

Instead, considering the following system:

$$ 1a + 2b = 2 \\ −3a + 4b = 5 $$

The goal of creating vectors here is that we want to be able to write the set of equations above as $aA + bB = C$ for some column vectors $A, B, C$. If you think about it in those terms, then it becomes more straightforward why you need to define:

$$ A=\begin{bmatrix}1 \\ -3\end{bmatrix} B=\begin{bmatrix}2 \\ 4\end{bmatrix} C=\begin{bmatrix}2 \\ 5\end{bmatrix} $$

Now we can re-write this system as: $aA + bB = C$.

$$ \begin{bmatrix}1a \\ -3a\end{bmatrix} + \begin{bmatrix}2b \\ 4b\end{bmatrix} = \begin{bmatrix}2 \\ 5\end{bmatrix} $$

It is not so much that these vectors represent vectors in the traditional plane, but rather as a concise way to represent the system of equations. Ultimately, people will begin to express these not as equations in vectors, but as augmented matrices, which nicely summarize the system of equations.

$$ \left[\begin{array}{cc|c} 1 & 2 & 2 \\ -3 & 4 & 5 \end{array}\right] $$

It is my opinion that suspending parallels to the $\mathbb{R}^2$ plane as soon as possible when studying linear algebra tends to be a worthwhile endeavour. It can be useful when drawing parallels, but ultimately hampers the ability to think more broadly about the concepts.

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    Yes, ok, that makes sense to me as I do understand the 'arithmetic' behind combining those vectors to come up with a solution to the system. – 2017-01-31
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    By the sounds of it the hiccup you are having is not as a result of the actual mathematics but rather the convention of working with abstract mathematical objects. Unfortunately, the example is given in 2-Dimensions which has the downside of being familiar in other aspects. Consider an example in say, 5 dimensions; at this point, vectors begin to lose their conventional description and can be thought of much more abstractly! – 2017-01-31
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    Upon further (and much closer) review of the text (thanks to both of your suggestions) it is apparent that Prof Strang was very careful to not label the axes in his 'vector drawing' as x and y - the way he intentionally did do so when he drew the traditional 'row' picture of two equations intersecting at a point in the traditional x and y plane. I'm still not fully clear on the 'warrant' brings us from equations to vectors, but the quick responses helped me overcome my 'block' on this, even though I do think there is much more for me to grasp on this detail. Thanks again, Luke – 2017-01-31
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    Happy to hear - my best recommendation would be to keep working through this. As you gain some familiarity you will quickly become intuitively familiar with these objects, much in the same way that you probably are with the real numbers! Best of luck!! – 2017-01-31
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    @Luke, while I understand you need to get through the course, an admonition for the long-term knowledge of the subject: the "matricial" way of teaching linear algebra has serious weaknesses. (1) difficulty telling vectors from coordinates with respect to a basis, (2) failure to clarify which aspects are inherent to finite dimension, and which aren't, and, above all, (3) failure to teach the *geometry* (i.e., the *invariant* properties) of linear spaces. P. Halmos's *Finite-dimensional vector spaces* and, as a supplement for speed, *Linear algebra problem book* are the only ones I recommend. – 2017-01-31
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    @avs, Thanks for the suggested books - I looked at both on Amazon, and they definitely come highly praised, and do seem to be what I'm really searching after. I've gone through a large amount of undergraduate math, but from an engineer's perspective, and am looking to fill it in with a more abstract treatment. Unfortunately, I can't dedicate full time to studying these subjects so need to do it on my own time using free resources like OCW, but am certainly after resources that will assist me in my quest to get "into" more abstract mathematics. Thanks again. – 2017-02-01