Can I know the if the vectors are a linear combination this way?

[The Student]

My last thread "Trying to Understand What a Matrix Represents" that I posted (still on the first page of threads) was because of the question below.

With no other information given, the question asks:

Can each vector in R4 be written as a linear combination of the columns of the matrix A below?

A =

1	3	0	3
-1	-1	-1	1
0	-4	2	-8
2	0	3	-1

This question is in my textbook, and I also found this question online from Pennsylvania State University

A is a matrix given in the following. Can each vector in R4 be written as a linear combination of the columns of the matrix A? Do the columns of A span R4? Explain your answer.   

1 3 0 3
−1 −1 −1 1
0 −4 2 −8
2 0 3 −1
   
Hint: this is a homework problem. Solution 0.4. You can use the method we discussed in class(Check your notes), or you can simply work on this given matrix, if it has a pivot at each row, then every vector is a linear combination of the columns of A(Why?), and these columns span R4. Reduce this matrix to echelon form, you will have a zero row at the bottom,so for some vectors in R4 it is not a linear combination of the column vectors. Of course they do not span R4. CAUTION: If you have to relate the given matrix with a linear sys- tem, it will be the COEFFICIENT matrix, NOT the augmented ma- trix. Thus you can not say anything about consistency or inconsis- tency to this matrix.

So I asked in my last thread if there was some kind of inherent meaning to just the numbers in a matrix. This question seems to imply that there is. So what vectors are they referring too?

Could the vectors that the question refers to be the columns? If so, in short is the question asking whether or not the columns are linear combinations of each other?

Finally, if my simplified version of the question is correct, couldn't we instantly tell that the answer is no because at least one column is not a multiple of another?

 

[pka]

My last thread "Trying to Understand What a Matrix Represents" that I posted (still on the first page of threads) was because of the question below.
With no other information given, the question asks:
Can each vector in R4 be written as a linear combination of the columns of the matrix A below?
A =

1	3	0	3
-1	-1	-1	1
0	-4	2	-8

Consider this caculation.

As you can see the determent is non-zero. That tells us that the matrix has an inverse.

Use the inverse to change the basis.

 

[mmm4444bot]

So I asked in my last thread if there was some kind of inherent meaning to just the numbers in a matrix.

We answered that question. It depends.

Math exercises may or may not correspond to some real-world meaning.

It's like being asked to find the derivative of -16x^2 + 50x + 2500. This quadratic polynomial has a real-world meaning, but one does not need to consider it, to complete the exercise.

The numbers in an augmented coefficient-matrix [n by m] represent a system of n equations in m variables.

What those equations represent is often unknown.

 

[The Student]

Consider this caculation.

As you can see the determent is non-zero. That tells us that the matrix has an inverse.

Use the inverse to change the basis.

(I did not get a very good mark in my linear algebra course, so I have decided to start from the beginning. At this point, we had only learnt the very basics which is roughly the first semester of a typical linear algebra first-year course. Determinants came in the second semester.)

It's not so much the procedure of finding the answer that I am having trouble with; it's understanding the question. It refers to vectors in R4. How do I know what these vectors are? Is there something that is implied by just having a 4 by 4 matrix?

 

[pka]

(I did not get a very good mark in my linear algebra course, so I have decided to start from the beginning. At this point, we had only learnt the very basics which is roughly the first semester of a typical linear algebra first-year course. Determinants came in the second semester.)
It's not so much the procedure of finding the answer that I am having trouble with; it's understanding the question. It refers to vectors in R4. How do I know what these vectors are? Is there something that is implied by just having a 4 by 4 matrix?

You should have written all of that upfront.
In my opinion, you are asking all the wrong questions. You should not worry about what mathematical concepts are but rather what they do for us? Not what they are but what they do. To understand that one must understand the operations.

Years ago I would begin linear algebra courses with "A matrix is a rectangular array of numerical data points" Then I would say "you must not worry about what that means, but MUST learn how the operations work".

If that is something that you can't do, then change you course of study. If you are one who must know what things concepts represent then mathematics is not for you.

 

[The Student]

You should have written all of that upfront.
In my opinion, you are asking all the wrong questions. You should not worry about what mathematical concepts are but rather what they do for us? Not what they are but what they do. To understand that one must understand the operations.

Years ago I would begin linear algebra courses with "A matrix is a rectangular array of numerical data points" Then I would say "you must not worry about what that means, but MUST learn how the operations work".

If that is something that you can't do, then change you course of study. If you are one who must know what things concepts represent then mathematics is not for you.

That is kind of a blind generalization; don't you think? How could it possibly hurt me to understand the philosophy of mathematics along with the mathematics? My interest is in both. Besides all of that, the question seems to be referring to vectors; what are these vectors that it is referring to? I don't see how this isn't a crucial question that I am asking.

 

[mmm4444bot]

Please post questions about the philosophy of mathematics (not to be confused with mathematical philosophy) on the Math Odds&Ends board. Thanks! :cool:

 

[The Student]

Please post questions about the philosophy of mathematics (not to be confused with mathematical philosophy) on the Math Odds&Ends board. Thanks! :cool:

I hate to keep bringing this up, so I will be more specific about what is confusing me about the question. The question refers to vectors. What do you think are these vectors that it is referring to?

I will make sure not to ask philosophical questions in this sub-forum.

 

[HallsofIvy]

You could determine "Can every vector in \(\displaystyle R^4\) can be written as a linear combination of the columns of matrix A" (thought of as vectors) by trying to do it!
Any vector in \(\displaystyle R^4\) is of the form (w, x, y, z) for some numbers w, x, y, and z and saying that vector can be written as a linear combination of the columns of A (though of as individual vectors) is the same as saying that there exist numbers, a, b, c, and d, such that
(w, x, y , z)= a(1, -1, 0, 2)+ b(3, -1, -4, 0)+ c(0, -1, 2, 3)+ d(3, 1, -8, -1). Of course, that is the same as the four simultaneous equations
a+ 3b+ 3d= w,
-a- b+ 2c+ d= x,
-4b+ 2c- 8d= y, and
2a+ 3c- d= z

Is it possible to solve those four equations for a, b, c, and d (as functions of w, x, y, and z)?


However, one of the things you may have learned is that such a system of equation will have a solution if and only if the determinant of the coefficients, which is just the determinant of your matrix, A, is NOT 0. So probably the easiest way to answer this question is to find the determinant of matrix A.

 

[The Student]

You could determine "Can every vector in \(\displaystyle R^4\) can be written as a linear combination of the columns of matrix A" (thought of as vectors) by trying to do it!
Any vector in \(\displaystyle R^4\) is of the form (w, x, y, z) for some numbers w, x, y, and z and saying that vector can be written as a linear combination of the columns of A (though of as individual vectors) is the same as saying that there exist numbers, a, b, c, and d, such that
(w, x, y , z)= a(1, -1, 0, 2)+ b(3, -1, -4, 0)+ c(0, -1, 2, 3)+ d(3, 1, -8, -1). Of course, that is the same as the four simultaneous equations
a+ 3b+ 3d= w,
-a- b+ 2c+ d= x,
-4b+ 2c- 8d= y, and
2a+ 3c- d= z

Is it possible to solve those four equations for a, b, c, and d (as functions of w, x, y, and z)?


However, one of the things you my have learned is that such a system of equation will have a solution if and only if the determinant of the coefficients, which is just the determinant of your matrix, A, is NOT 0. So probably the easiest way to answer this question is to find the determinant of matrix A.

Thank-you, it just clicked. I misunderstood the question.

 

[The Student]

Thanks everyone for your patience and help!