Linear function standard basis.

[Matriculator]

Does anyone know what L is? I'm trying to see if I could find videos on it on YouTube.

On the first question this is what I think- [a;b] is a vector by the way:

1) [2;1]c₁+[7;4]c₂=[1;0]
[2;1]c₁+[7;4]c₂=[0;1]

I could have also combined those two by having the linear combination equal to a size 2 identity matrix, right?! Is this correct?

2) This is where I'm lost. I know how to switch between a standard basis and a given basis. I'm not exactly sure of the nature of the equation being asked or what it even is, including what L is.

 

[Mugged]

Let me take a guess at L. From the first condition, it takes an x and transforms to a y. And from the second condition, it takes a y and transforms to a -x.

L = \begin{bmatrix}
0 & -1 \\
1 & 0 \\
\end{bmatrix}

you can check that this L satisfies your mapping constraints.

 

[vela]

Does anyone know what L is? I'm trying to see if I could find videos on it on YouTube.

What do you mean? The problem statement tells you what L is. It's a linear function between R² and R².

On the first question this is what I think- [a;b] is a vector by the way:

1) [2;1]c₁+[7;4]c₂=[1;0]
[2;1]c₁+[7;4]c₂=[0;1]

Strictly speaking, what you wrote has no solution because there are no $c_1$ and $c_2$ that satisfy both equation at once. Assuming you're just being sloppy, I'd say you have the right idea.

I could have also combined those two by having the linear combination equal to a size 2 identity matrix, right?! Is this correct?

Not really. How can a linear combination of column vectors produce a 2x2 matrix?

2) This is where I'm lost. I know how to switch between a standard basis and a given basis. I'm not exactly sure of the nature of the equation being asked or what it even is, including what L is.

What do you mean when you say you "know how to switch between a standard basis and a given basis"?