Answer to Question #206992 in Linear Algebra for Rohan Kumar

Question

Let V be the vector space of all 2×2 matrices over the field R. let W₁ = {$\begin{bmatrix} x & -x \\ y & z \end{bmatrix}$| x,y,z€R} and W₂= {$\begin{bmatrix} a & b \\ -a & c \end{bmatrix}$| a,b,c€R}. What is the dimensions of W₁+W₂ and W₁ intersection W₂ as well?

Answer 1

$W_1 =\{\begin{bmatrix} x & -x \\ y & z \end{bmatrix}| x,y,z,\in R\}$

Basis $W_1$ , for example,

$E_1=\begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, E_2=\begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, E_3=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$

$dimW_1=3$

$W_2 =\{\begin{bmatrix} a & b \\ -a & c \end{bmatrix}| a,b,c,\in R\}$

Basis $W_2$ , for example,

$E'_1=\begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix}, E'_2=\begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, E'_3=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$

$dimW_2=3$

$W_1+W_2:\\ a_1E_1+a_2E_2+a_3E_3+a_4E'_1+a_5E'_2=0\\ \begin{bmatrix} a_1+a_4 & -a_1+a_5\\ a_2-a_4 & a_3 \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

$a_1+a_4=0\\ -a_1+a_5=0\\ a_2-a_4=0\\ a_3=0$

$a_4=-a_1\\ a_5=a_1\\ a_2=a_4=-a_1\\ a_3=0$

$dim(W_1+W_2)=5\\ dim(W_1\cap W_2)=dimW_1+dimW_2-\\ -dim(W_1+W_2)=3+3-5=1$