Trying to Understand What a Matrix Represents

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The Student

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I have an idea of what the rows of Ax= b means, but I don't understand what just the matrix A means. For example, a 3 by 3 matrix seems to mean that each column is a vector in 3 dimensions. Do the rows mean the same thing, or are they just "information" about the columns?
 
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I would NOT say that each column of a matrix is a vector, although I can see how that might be useful in some circumstances. And, of course, you can think of each row as a vector in exactly the same way. A simple way of thinking about matrix products is that, to Multiply AB, you take the dot product of each "row vector" in A with each "column vector" in B. That is, the "ij" entry in AB is the dot product of the ith row in A with the jth column in B.
 
HallsofIvy said:
I would NOT say that each column of a matrix is a vector, although I can see how that might be useful in some circumstances. And, of course, you can think of each row as a vector in exactly the same way. A simple way of thinking about matrix products is that, to Multiply AB, you take the dot product of each "row vector" in A with each "column vector" in B. That is, the "ij" entry in AB is the dot product of the ith row in A with the jth column in B.
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But before A gets multiplied by anything, what does the matrix A represent? In other words, what do the numbers, in say a 3 by 3 matrix, mean?
 
The Student said:
But before A gets multiplied by anything, what does the matrix A represent? In other words, what do the numbers, in say a 3 by 3 matrix, mean?
Click to expand...

Matrix is just a "shorthand" method in mathematics - it means different things under different situation.

In case you are trying to solve a set of linear equations, the augmented matrix is the shorthand way of letting the "solver" (be it human or computer) have all the information regarding that problem.

In mechanics, the state of stress at a point - under plane stress condition - is described with a 3 X 3 matrix ( so is state of stress).

The generalized compliance matrix (containing all the moduli for mechanics) is a 4 dimensional matrix (3 X 3 X 3 X 3) - which is simplified to a 2-dimensional 6 X 6 matrix implementing certain physical conditions.

So you see matrix is just a short-hand way of transferring certain information. Nature of that information changes with change in problem.
 
Subhotosh Khan said:
Matrix is just a "shorthand" method in mathematics - it means different things under different situation.

In case you are trying to solve a set of linear equations, the augmented matrix is the shorthand way of letting the "solver" (be it human or computer) have all the information regarding that problem.

In mechanics, the state of stress at a point - under plane stress condition - is described with a 3 X 3 matrix ( so is state of stress).

The generalized compliance matrix (containing all the moduli for mechanics) is a 4 dimensional matrix (3 X 3 X 3 X 3) - which is simplified to a 2-dimensional 6 X 6 matrix implementing certain physical conditions.

So you see matrix is just a short-hand way of transferring certain information. Nature of that information changes with change in problem.
Click to expand...

But don't the numbers in the matrix have some kind of meaning or significance on their own? Do they represent points for different dimensions or weights for different dimensions, or anything like that?
 
Exactly what kind of course are you taking? Is it just "matrices" or is this part of a Linear Algebra course, dealing with "vector spaces"? A "linear transformation is linear function from one vector space to another or from one vector space to itself. A matrix is a set of numbers representing a linear function in given bases for the vector spaces in the same way that, given a basis, i, j, k, we can write the vector ai+ bj+ ck as \(\displaystyle \begin{bmatrix}a \\ b \\ c \end{bmatrix}\).

Notice that if we apply the matrix \(\displaystyle \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\) to the vector \(\displaystyle \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}\), the result is the first column, \(\displaystyle \begin{bmatrix}a \\ d \\ g\end{bmatrix}\). Similarly for the vectors \(\displaystyle \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\) and \(\displaystyle \begin{bmatrix}0 \\ 0\\ 1\end{bmatrix}\)- they give the second and third columns. Any linear transformation, from vector space U to vector space V, takes all of U to some subspace of V (perhaps all of V) and takes basis vectors of U to vectors that span V. So, if we have a basis for U and write the linear transformation as a matrix, the columns of the matrix are the vectors, in v, that those basis vectors are taken to. The individual numbers are the coefficients of the basis vectors for V in those vectors.
 
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