In order to verify if a given matrix is a Linear Transformation

[number0]

Homework Statement

I know I am suppose to show that the matrix is closed under addition and multiplication properties, but is it POSSIBLE for me to show that the vector columns can be spanned in the given R^m (assume that the Linear Transformation happens from R^n -> R^m) ?

For example,

[x + y]
[ x ].

This is a linear transformation becausex[1] + y[1]
[1] [0] =span([1] [1])
[1], [0]Can I do this?

Homework Equations

The Attempt at a Solution

 

[Mark44]

What you showed was difficult to comprehend, so I added code tags to preserve your formatting.

Homework Statement

I know I am suppose to show that the matrix is closed under addition and multiplication properties, but is it POSSIBLE for me to show that the vector columns can be spanned in the given R^m (assume that the Linear Transformation happens from R^n -> R^m) ?

To clarify your question, yes it's possible to show that the vector columns span R^m, but why do you want to do this?

If you're supposed to show that the matrix represents a linear transformation, show that L(x + y) = L(x) + L(y), and that L(cx) = cL(x), where L is the linear transformation that represents your matrix, x and y are arbitrary vectors in Rⁿ, and c is a scalar.

For example,

[x + y]
[  x   ].

This is a linear transformation because

x[1]   + y[1]
  [1]       [0]   =

span([1]  [1])
       [1], [0]

Can I do this?

Homework Equations

The Attempt at a Solution