Find the orthogonal projection

[yevi]

Homework Statement

My questions is this:
How to find the orthogonal projection of vector y= (7,-4,-1,2) on null space
N(A)

Where A is a matrix
A =

$$\left(\begin{array}{cccc}2&1&1&3\\3&2&2&1\\1&2&2&-9\end{array}\right)$$

Homework Equations

$$A^TA\overline{x}=A^T\overline{y}$$

The Attempt at a Solution

First I found the Null space of matrix A:
A =

$$\left(\begin{array}{cc}0&-5\\-1&7\\1&0\\0&1\end{array}\right)$$

Then, I applied he formula from aboce:

A^TA =
2 -7
-7 75

A^Ty= (3,-61)

after that built an equation to find x:

$$\left(\begin{array}{cc}2&-7\\-7&75\end{array}\right) \left(\begin{array}{c}X1\\X2\end{array}\right) = \left(\begin{array}{c}3\\-61\end{array}\right)$$

x1 = -2 , x2=-1
P(x) = (5,-5,-2,-1)

But the answer is:
3/2(0,-1,1,0)

What is wrong?

 

[mighty2000]

it is important to first understand the concepts behind the problem before attempting to solve it. The null space of a matrix A is the set of all vectors that, when multiplied by A, result in the zero vector. The orthogonal projection of a vector y onto the null space of A is the vector that is closest to y and lies in the null space of A. This means that the orthogonal projection of y onto the null space of A can be represented as Ax, where x is a vector in the null space of A.

To find the orthogonal projection of vector y onto the null space of A, we can use the formula P = A(A^TA)^-1A^Ty, where P is the orthogonal projection and (A^TA)^-1 is the inverse of the matrix A^TA. This formula can be derived from the projection matrix formula P = A(A^TA)^-1A^T.

Using this formula, we can calculate the orthogonal projection of y onto the null space of A as follows:

1. Find the null space of A: From the given matrix A, we can see that the null space of A is spanned by the vectors (0,-1,1,0) and (0,1,0,1). This means that any vector in the null space of A can be represented as a linear combination of these two vectors.

2. Calculate A^TA: We can calculate A^TA as follows:

A^TA =
2 -7
-7 75

3. Calculate (A^TA)^-1: The inverse of A^TA can be calculated using matrix inversion techniques or by using a calculator. In this case, we get (A^TA)^-1 = (1/75) * (75 7; 7 2).

4. Calculate A^Ty: We can calculate A^Ty as follows:

A^Ty =
3
-61

5. Calculate P: Now, we can calculate the orthogonal projection as follows:

P = A(A^TA)^-1A^Ty
=
\left(\begin{array}{cc}0&-1\\1&0\\1&1\\0&1\end{array}\right) * \left(\begin{array}{cc}1/75&-7/75\\-7/75&2/75\end{array}\right