Finding the basis for a subspace in vectorspace

[Rhabdovirus]

Homework Statement

Find the basis for the subspace S of the vector space V. Specify the dimension of S.

S={a a+d} where a,d are elements of R and V= M_2x2
{a+d d }

Homework Equations

I guess I know the standard basis for M_2x2 are the [(10 00) (01 00) (00 10) (00 01)] matrices

The Attempt at a Solution

I don't very well know where to start, I can try using an augmented matrix setting c₁S = [(ef) (g h)] and finding a consistent linear system...


I'm sorry I probably didn't format my matrices correctly, I'm going to start teaching myself LaTeX over winter break.

 

[Mark44]

Homework Statement

Find the basis for the subspace S of the vector space V. Specify the dimension of S.

S={a a+d} where a,d are elements of R and V= M_2x2
{a+d d }

Homework Equations

I guess I know the standard basis for M_2x2 are the [(10 00) (01 00) (00 10) (00 01)] matrices

The Attempt at a Solution

I don't very well know where to start, I can try using an augmented matrix setting c₁S = [(ef) (g h)] and finding a consistent linear system...


I'm sorry I probably didn't format my matrices correctly, I'm going to start teaching myself LaTeX over winter break.

For a given matrix in V, you have two free variables, a and d. Does that suggest how many vectors (matrices in this case) make up a basis?

 

[Rhabdovirus]

to me, it sounds like I have 2 matrices... a 2 equations, 2 unknowns kind of deal. would that then be my basis? I suppose one of my problems is being lost in this chapter

 

[Mark44]

Yep, I get two matrics. You can pick a and d, and these values determine the off-diagonal entries. Try a = 1, d = 0, and a = 0, d = 1.

 

[Rasalhague]

I'm sorry I probably didn't format my matrices correctly, I'm going to start teaching myself LaTeX over winter break.

For the basic stuff, a handy short cut is to use an online LaTeX equation editor such as this, then copy & paste. The more you use it, the more you'll learn without even trying.

 

[Rhabdovirus]

For the basic stuff, a handy short cut is to use an online LaTeX equation editor such as this, then copy & paste. The more you use it, the more you'll learn without even trying.

thanks, I actually have a primer for it (i think it's the not so short intro to latex) that one of my prof's recommended, but I'll definitely use that when I need more help!

And thanks a lot to you Mark!

 

[Mark44]

A lot of the LaTeX stuff is pretty easy. Here are some examples, with how the script would look on the left, and how it renders on the right. (The extra spaces in the tex and /tex tags keep the script from rendering.)

Exponents
[ tex]e^{2t}[ /tex] e^{2t}

Subscripts
[ tex]a_i + a_{i + 1}[ /tex] a_i + a_{i + 1}

Fractions
[ tex]\frac{2x^2 + 3x}{x^3 - 1}[ /tex] \frac{2x^2 + 3x}{x^3 - 1}

Integrals
[ tex]\int \sqrt{1 - x^2}dx[ /tex] \int \sqrt{1 - x^2}dx

[ tex]\int_0^{1} \sqrt{1 - x^2}dx[ /tex] \int_0^{1} \sqrt{1 - x^2}dx

Limits
[ tex]\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[ /tex] \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

Matrices
These are a little more involved, and there are several ways of doing them. This is the way I'm most comfortable with.
[ tex]\left[ \begin{array}{c c} a& a+d\\a + d & d\end{array}\right] [ /tex]
\left[ \begin{array}{c c} a& a+d\\a + d & d\end{array}\right]