C21

C21 Find the dimension of the subspace \(W = \{\begin{bmatrix}a + b\\ a + c\\ a + d\\ d\\ \end{bmatrix} | a,b,c,d \in \mathbb{C} \}\) of \(\mathbb{C}^4\)

At first glance, we can assume that since it’s of \(\mathbb{C}^4\), then the dimension should be 4, but we need to make sure by checking for any linear combinations. We can see that the matrix is actually of the form \(\begin{bmatrix} 1a + 1b + 0c + 0d\\ 1a + 0b + 1c + 0d\\ 1a + 0b + 0c + 1d\\ 0a + 0b + 0c + 1d\\ \end{bmatrix}\), thus we can factor out the scalars to give us \(a \begin{bmatrix}1\\ 1\\ 1\\ 0\\ \end{bmatrix} + b \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix} + c \begin{bmatrix} 0\\ 1\\ 0\\ 0\\ \end{bmatrix} + d\begin{bmatrix}0\\ 0\\ 1\\ 1\\ \end{bmatrix}\) and then finally reduce again to give us four vectors of \(\{\begin{bmatrix} 1\\ 1\\ 1\\ 0\\ \end{bmatrix}, \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1\\ 1\\ \end{bmatrix}\}\)

We can see that this reduced form can be turned into an identity matrix. Subtracting R4 from R3, R3 from R1, and R3 from R2 will give us this \(\{\begin{bmatrix} 0\\ 0\\ 1\\ 0\\ \end{bmatrix}, \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 0\\ 1\\ \end{bmatrix}\}\), and then by rearranging the rows we’re left with the identity matrix \(\{\begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1\\ 0\\ \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 0\\ 1\\ \end{bmatrix}\}\)

This means the vectors are linearly independent and forms a basis of \(W\), so our first guess of 4 is correct.