a \(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\) + b \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\) = \(\begin{bmatrix} 4 \\ 3 \end{bmatrix}\)

\(\begin{bmatrix} 2 & 1 & 4 \\ 1 & 1 & 3 \end{bmatrix}\)

Row reduce:

library(pracma)
## Warning: package 'pracma' was built under R version 4.3.2
A <- matrix(c(2, 1, 4, 1, 1, 3), 2, 3, byrow = TRUE)
rref(A)
##      [,1] [,2] [,3]
## [1,]    1    0    1
## [2,]    0    1    2

Now, you know that a equals 1 and b equals 2 because the third column in the matrix shows you the answer. Plug in to the equation, and you will get the same result.

1 * T\(\begin{bmatrix} 2 \\ 1 \end{bmatrix}\) + 2 * T\(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\) = T\(\begin{bmatrix} 4 \\ 3 \end{bmatrix}\)

1 * \(\begin{bmatrix} 3 \\ 4 \end{bmatrix}\) + 2 * \(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\) = \(\begin{bmatrix} 1 \\ 8 \end{bmatrix}\)