Finding a row vector and calculating the trace of a matrix

In summary: Thanks for trying to help.In summary, Homework statement (i) states that a function, R2 → R2, is defined by (3x−4y, x−2y) and v = (a,b). The vector v is found by solving f (v) = (4, 6) and a+b is found to be -15. Homework statement (ii) states that f : R3 → R3 is a projection onto the xz-plane and that E is the basis for R3. Matrix A of f is found to be trace(A) and is calculated to be -0.
  • #1
hen93
2
0
Hi there, I've been having trouble with 2 algebra questions, I was hoping someone here could give me a hand.

Homework Statement


(i) Consider the function R2 → R2 defined by f (x, y) = (3x − 4y, x − 2y). Let
v = (a, b) be the vector such that f (v) = (4, 6).
Find the vector v and hence calculate a + b.

(ii) Let f : R3 → R3 be a projection onto the xz-plane. Choose your favourite
basis E for R3 and calculate the matrix A of f with respect to E.
Calculate trace(A)

Homework Equations


N/A.

The Attempt at a Solution



(i) I am positive that this must be incorrect but i took that 3a - 4b =4 and a -2b = 6. Solving for a+b =-15.

(ii)I understand that the trace of a matrix is the sum of all the diagonal entries starting from the top left corner, but the phrasing question has left me clueless.

Any help would be greatly appreciated.
Thank you.
 
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  • #2
hen93 said:
Hi there, I've been having trouble with 2 algebra questions, I was hoping someone here could give me a hand.

Homework Statement


(i) Consider the function R2 → R2 defined by f (x, y) = (3x − 4y, x − 2y). Let
v = (a, b) be the vector such that f (v) = (4, 6).
Find the vector v and hence calculate a + b.

(ii) Let f : R3 → R3 be a projection onto the xz-plane. Choose your favourite
basis E for R3 and calculate the matrix A of f with respect to E.
Calculate trace(A)


Homework Equations


N/A.


The Attempt at a Solution



(i) I am positive that this must be incorrect but i took that 3a - 4b =4 and a -2b = 6. Solving for a+b =-15.
You did it correctly. Why do you think it's wrong?

(ii)I understand that the trace of a matrix is the sum of all the diagonal entries starting from the top left corner, but the phrasing question has left me clueless.
What specifically is confusing you? Do you know what the difference between a linear transformation and the matrix that represents it is?
 
  • #3
vela said:
You did it correctly. Why do you think it's wrong?

What specifically is confusing you? Do you know what the difference between a linear transformation and the matrix that represents it is?
Sorry, I just thought that was to simple to be correct.
I think that I do, just that without any numbers it does not make any sense to me.
 
  • #4
Well, coming up with the numbers is the whole problem. So why not start as suggested and pick your favorite basis for R3. Do you know how to find the matrix once you've chosen a basis? If not, that's what you need to look into.
 

FAQ: Finding a row vector and calculating the trace of a matrix

What is a row vector?

A row vector is a one-dimensional array of numbers, often represented horizontally, that contains the elements of a single row in a matrix. It is used to represent data in linear algebra and can be manipulated using various mathematical operations.

How do I find a row vector in a matrix?

To find a row vector in a matrix, you can simply select a row from the matrix and write it out as a one-dimensional array. For example, if you have a matrix A = [1 2 3; 4 5 6; 7 8 9], the second row would be represented as [4 5 6].

What is the purpose of a row vector in matrix operations?

A row vector is often used in matrix operations because it allows for efficient manipulation of data. By breaking down a matrix into its individual rows, it becomes easier to perform operations such as addition, subtraction, and multiplication.

How do I calculate the trace of a matrix?

The trace of a matrix is the sum of the elements on the main diagonal (top left to bottom right) of the matrix. To calculate the trace, simply add up all the numbers on the main diagonal. For example, if you have a matrix A = [1 2 3; 4 5 6; 7 8 9], the trace would be 1 + 5 + 9 = 15.

Why is the trace of a matrix important in linear algebra?

The trace of a matrix is important in linear algebra because it is a measure of the sum of eigenvalues of a matrix. It is also used in various mathematical calculations, such as finding the determinant and solving systems of linear equations. Additionally, the trace has applications in physics, engineering, and other fields of science.

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