If I have a linear combination of two numbers $\lambda_1y_1+\lambda_2y_2$, I can represent it as $\mathbf y \cdot\vec \lambda$, i.e., an inner product of two vectors.
If I have a linear combination of two vectors $\lambda_1\mathbf y_1+\lambda_2 \mathbf y_2$, I can represent it as matrix-vector product $\mathbf Y\vec \lambda$.
How can I do something similar for linear combination of two matrices $\lambda_1 Y_1+\lambda_2 Y_2$?
In the language of Linear Algebra, the $y$'s are looked upon as vectors in some vector space, and what you do is writing those vectors' coordinate representation as columns of a matrix.
Hence this requires fixing a coordinate system (i.e. a basis) for the space first.
In your first example, the numbers can in fact be looked upon as vectors in the $1$-dimensional space, which is why each column's height in $\mathbf{y}$ is $1$.
In your third example, matrices (say $M_{n \times m}(\mathbb{R})$) can also be viewed as a $nm$-dimensional space, but writing linear combinations in a similar way requires writing each matrix as a column vector. For example,
$\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ \end{bmatrix}$ for $\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}$.