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.