Definition of a $\mathbb{R}^d$-valued Gaussian process

Consider the following definitions from my lecture notes:

A $$\mathbb{R}^d$$-valued random variable $$X =(X_1, \dots, X_d)$$ is called $$d$$-dim Gaussian if every linear combination of its coordinates is (one dimensional) normal distributed.

A $$\mathbb{R}^d$$-valued stochastic process $$(X_t)$$ is called Gaussian if all its finite dimensional distributions are Gaussian.

I have some trouble with the dimensions: So to show that a $$\mathbb{R}^d$$-valued process $$(X_t)$$ is Gaussian, I pick some arbitrary times $$t_1, \dots, t_n$$ and consider the $$(d \times n)$$-valued (?) random variable $$(X_{t_1}, \dots , X_{t_n})$$. But then a linear combination of $$(X_{t_1}, \dots , X_{t_n})$$ won't be one dimensional?! So I can not use the given definition. Where is my mistake?

Yes, the definition is a little misleading. You have to think of your $$d\times n$$-valued random variable as a vector of length $$d\cdot n$$, not of a matrix, so just write all columns of the matrix one below the other.
I'm not sure if the definition is exact or where you'd like to use it. But by marginalization property of Gaussian random vectors, coordinates of $$X$$ are also Gaussian random variables (one dimensional). Also, Gaussian random variables remain Gaussian under linear (affine) transformations. So any linear combinations of coordinates of $$X$$ will be Gaussian. Hope this helps.