- #1

archaic

- 688

- 214

- Homework Statement
- Find a matrix ##A## such that its null space is ##\mathrm{span}(v_1,v_2)##, where ##v_i\in\mathcal{M}_{41}##.

- Relevant Equations
- rank + nullity = number of columns

I have solved the exercise, so I'm not giving the vectors explicitly. I just want to know if there is a quicker way than mine.

We know that ##A## must have ##4## columns and ##4## lines, and we also know that its nullity is ##2##, thus its rank is ##2##.

I took the simplest matrix that can have a rank ##2##, namely ##L_1 = (1,a,b,c)##, ##L_2=(0,1,d,e)##, and the rest is zero.

Then, I multiplied by ##tv_1+sv_2##, factorized ##t## and ##s## in each line, and made their coefficients ##0##.

The last step was to solve for ##a,\,b,\,c,\,d## and ##e## with the equations I put. ##c## is a free variable, so I took a convenient value.

We know that ##A## must have ##4## columns and ##4## lines, and we also know that its nullity is ##2##, thus its rank is ##2##.

I took the simplest matrix that can have a rank ##2##, namely ##L_1 = (1,a,b,c)##, ##L_2=(0,1,d,e)##, and the rest is zero.

Then, I multiplied by ##tv_1+sv_2##, factorized ##t## and ##s## in each line, and made their coefficients ##0##.

The last step was to solve for ##a,\,b,\,c,\,d## and ##e## with the equations I put. ##c## is a free variable, so I took a convenient value.