Question

Why are we able to think of column matrices as vectors? For instance, X(1,2,3) (assume the column matrix was vertical). If I wanted to draw that vector in R3, that would be 1 in the i hat direction, 2 in the j hat direction and 3 in the k hat direction. Why are we able to draw the column matrix as that kind of vector, especially if the scalars 1, 2 and 3 are all the coefficients of the x variable in three separate equation?

Answer 1

Hi I had the same question. However I looked at it this way
Suppose [1 2] is one Col in a matrix and [3 4] the other, then in [x y] I thought that 'x' would be the scaling factor of [1 2] and 'y' would be the scaling component factor of [3 4] to obtain a soon.
Is this a correct way of looking it ? Could anybody comment please ?

Answer 2

Isn't this just built into the definition of matrix multiplication? If I'm given an equation

3x+2y+6z = 4

I can look at the RHS as either

(3 2 6)( x or as (x y z) (3
y 2
z) 6)

Another way to think about it is to note that any set of three numbers can be thought of as a vector in R3. (Whether or not it's a useful way to think about it depends on the context in which they arose.)

Answer 3

Thank you. Things seem a lot clearer now. However I had a small clarification in this form
(3 2 6)(x
Y
z)

If (x y z) is considered as a vector in R3, it should have a direction- does it mean that axes are vectors that intersect at origin ? And (3 2 6) are scaling factors of the axes ?

Answer 4

The axes are (1 0 0) , (0 10) , and (0 0 1). (all transposed.)