Matrix, making R2 to R3, finding standard matrix A where did i mess up?

In summary, to convert a vector in R2 to R3 in matrix form, a third component (z-value) must be added to the vector by adding a row of zeros. This vector can then be multiplied by a 3x2 matrix with the first two columns representing the original vector and the third column representing the added zeros. The standard matrix for this conversion is a 3x2 matrix with the first two columns representing the original vector and the third column representing a row of zeros. To find the standard matrix A for a given transformation, the transformation must be applied to the standard basis vectors and the resulting vectors will form the columns of A. Some common mistakes when converting from R2 to R3 include forgetting to add the third
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mr_coffee
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Hello everyone I'm suppose to find the standard matrix A of T, R^2 -> R^3; I have all the work here, any help would be great:
http://img138.imageshack.us/img138/6602/lastscan2rm.opt.jpg [Broken]
 
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1. How do I convert a vector in R2 to R3 in matrix form?

To convert a vector in R2 to R3, you will need to add a third component (z-value) to the vector. This can be done by adding a row of zeros to the original vector. Then, multiply the vector by a 3x2 matrix with the first two columns representing the original vector and the third column representing the added zeros. This will give you a vector in R3 in matrix form.

2. What is the standard matrix for converting a vector in R2 to R3?

The standard matrix for converting a vector in R2 to R3 is a 3x2 matrix with the first two columns representing the original vector and the third column representing a row of zeros. This matrix can then be multiplied with the vector to convert it into R3 in matrix form.

3. How do I find the standard matrix A for a given transformation?

To find the standard matrix A for a given transformation, you will need to apply the transformation to the standard basis vectors (i and j) in R2 and observe the resulting vectors. The columns of A will be the resulting vectors in matrix form. For example, if the transformation is T(x,y) = (2x, y+1), the resulting vectors would be (2,0) and (0,1), making the standard matrix A = [2 0; 0 1].

4. What are some common mistakes when converting from R2 to R3?

One common mistake is forgetting to add the third component (z-value) to the vector when converting from R2 to R3. Another mistake is mixing up the order of the components when constructing the standard matrix. It is important to remember that the first two columns represent the original vector in R2 and the third column represents the added zeros.

5. How can I check if I have correctly converted a vector from R2 to R3?

To check if you have correctly converted a vector from R2 to R3, you can multiply the vector by the standard matrix A and see if the resulting vector matches the converted vector in R3. You can also plot the original vector and the converted vector in a 3D coordinate system to visually check if they are equivalent.

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