Linear function standard basis.

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

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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.

Matriculator said:
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"?

What is a linear function?

A linear function is a mathematical function that can be represented by a straight line on a graph. It follows the general form y = mx + b, where m is the slope of the line and b is the y-intercept.

What is the standard basis for a linear function?

The standard basis for a linear function is the set of basis vectors that serve as a standard or reference point for representing other vectors in the same space. In a two-dimensional coordinate system, the standard basis consists of the unit vectors i = (1,0) and j = (0,1).

What is the relationship between linear functions and the standard basis?

Linear functions can be expressed in terms of the standard basis vectors. For example, the linear function f(x) = 3x + 2 can be rewritten as f(x) = 3i + 2j, where i and j are the standard basis vectors. This allows for easier comparison and manipulation of linear functions.

How do you graph a linear function in relation to the standard basis?

To graph a linear function, you can plot points using the standard basis vectors i and j. For example, if a linear function has a slope of 2, you can plot points by starting at the origin and moving 2 units to the right (using i) and 2 units up (using j). This will create a straight line that represents the function.

What is the significance of the standard basis in linear algebra?

The standard basis is an important concept in linear algebra because it allows for the representation and manipulation of vectors in a standard or reference frame. It also helps in visualizing and understanding linear transformations, which are fundamental to linear algebra.

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