Matrix vector product and linear transformation proof

[nobahar]

Homework Statement

Hello!
Prove:
A(\vec{a}+\vec{b}) = A\vec{a} + A\vec{b}
Where A is a matrix and T (in the following section) is a transformation.

Homework Equations

T(\vec{a}) + T(\vec{b}) = T(\vec{a}+\vec{b})
T(\vec{a}) = A\vec{a}
T(\vec{b}) = A\vec{b}

The Attempt at a Solution

If \vec{a}+\vec{b} = \vec{c}
T(\vec{a}+\vec{b}) = T(\vec{c}) = Ac = A(\vec{a}+\vec{b})
T(\vec{a}+\vec{b}) = A(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} + A\vec{b}

Is this a sufficient proof? I can do it the more arduous way, but I think this is a proof, isn’t it?
Any help appreciated.

 

[Mark44]

Homework Statement

Hello!
Prove:
A(\vec{a}+\vec{b}) = A\vec{a} + A\vec{b}
Where A is a matrix and T (in the following section) is a transformation.

Homework Equations

T(\vec{a}) + T(\vec{b}) = T(\vec{a}+\vec{b})
T(\vec{a}) = A\vec{a}
T(\vec{b}) = A\vec{b}

The Attempt at a Solution

If \vec{a}+\vec{b} = \vec{c}
T(\vec{a}+\vec{b}) = T(\vec{c}) = Ac = A(\vec{a}+\vec{b})
T(\vec{a}+\vec{b}) = A(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} + A\vec{b}

Is this a sufficient proof? I can do it the more arduous way, but I think this is a proof, isn’t it?
Any help appreciated.

I assume you are using the properties of a linear transformation. I would do it this way.

A(\vec{a} + \vec{b}) = T(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} +A\vec{b}

 

[nobahar]

I assume you are using the properties of a linear transformation. I would do it this way.

A(\vec{a} + \vec{b}) = T(\vec{a}+\vec{b}) = T(\vec{a}) + T(\vec{b}) = A\vec{a} +A\vec{b}

Thanks Mark44.