Why is this a unit square?

[The Student]

Why is this matrix a unit square?

1	0
0	1

It seems like it would just be a point at (1, 0), and a point at (0, 1). How does the square get filled up; or does it get filled up?

 

[pka]

Why is this matrix a unit square?

1	0
0	1

It seems like it would just be a point at (1, 0), and a point at (0, 1). How does the square get filled up; or does it get filled up?

Are you mixing up apples and oranges?

1) \(\displaystyle \Re^2\) is a vector space with a basis of \(\displaystyle (1,0)~\&~(0,1)\).

2) \(\displaystyle \mathfrak{M}_2\) is the vector space of \(\displaystyle 2\times 2\) matrices over some field.
That space has a basis of four basic elements.
In this space the operational identity is \(\displaystyle \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right)\)

Apples are not oranges.

 

[The Student]

Are you mixing up apples and oranges?

1) \(\displaystyle \Re^2\) is a vector space with a basis of \(\displaystyle (1,0)~\&~(0,1)\).

2) \(\displaystyle \mathfrak{M}_2\) is the vector space of \(\displaystyle 2\times 2\) matrices over some field.
That space has a basis of four basic elements.
In this space the operational identity is \(\displaystyle \left( {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right)\)

Apples are not oranges.

I haven't learnt about basis' yet, and I don't know what the symbol in 2) is. I guess that I am still confused about what the matrices mean and why they mean what they mean. A few weeks ago, I was told that a matrix is just some data points and that there was no geometrical description of just a matrix. Now it seems that the matrix in my original post has a geometrical description/meaning. Please help.

 

[HallsofIvy]

I haven't learnt about basis' yet, and I don't know what the symbol in 2) is. I guess that I am still confused about what the matrices mean and why they mean what they mean. A few weeks ago, I was told that a matrix is just some data points and that there was no geometrical description of just a matrix. Now it seems that the matrix in my original post has a geometrical description/meaning. Please help.

Where in the world did you hear that? An "array" of numbers can be thought of as "just some data points" but the whole point of "matrices" is that we have specific operations, multiplication, addition, and scalar multiplication. A matrix is NOT a "geometric object" but can be interpreted in several different geometric ways in specific applications.

Now, what do you mean by "unit square"? What geometric interpretation are you giving to this matrix or where did you see this matrix referred to as a "unit square". What you give is certainly a "square matrix" because it has the same number of rows and columns, it is the multiplicative "identity", and, like any multiplicative identity, it is the square of itself (\(\displaystyle I^2= I\)) so a "perfect square", but I would not call it a "unit square".

 

[The Student]

Where in the world did you hear that? An "array" of numbers can be thought of as "just some data points" but the whole point of "matrices" is that we have specific operations, multiplication, addition, and scalar multiplication. A matrix is NOT a "geometric object" but can be interpreted in several different geometric ways in specific applications.

Now, what do you mean by "unit square"? What geometric interpretation are you giving to this matrix or where did you see this matrix referred to as a "unit square". What you give is certainly a "square matrix" because it has the same number of rows and columns, it is the multiplicative "identity", and, like any multiplicative identity, it is the square of itself (\(\displaystyle I^2= I\)) so a "perfect square", but I would not call it a "unit square".

My textbook shows the matrix that I have in my original post as having a geometrical description of what it called a "unit square". The picture of the geometrical description is essentially a shaded in area of a square with the corners: (0, 0), (1, 0), (1, 1), (0, 1) on a Cartesian coordinate system. This is in the "Linear Transformations" chapter if that helps to put it in context.

 

[The Student]

It is certainly NOT a unit square; seems to be related to the coordinates
of a unit square located at (0,0), (0,1), (1,1), (1,0).

I should have been more clear. I want to know why its geometrical description is a unit square.

In my textbook, the unit square is shaded in. Does this mean that the matrix in my original post geometrically describes all points in the unit square or just the points (0,0), (0,1), (1,1), (1,0)?