YouLet v1=[-1 4 0 0 0 -1], v2=[-5 0 2 -1 0 -2], v3=[5 0 0 3 3 5]Use the Gram-Schmidt procedure to produce an orthogonal set with the same span. what is u1, <u1, v2>, <u1, u1>, u2, <u1, v3>, <u2, v3>, <u2, u2>, and u3.

The Gram-Schmidt procedure is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space R^n. The procedure involves taking a set of vectors and producing an orthogonal set with the same span.

Let's start with the vectors v1, v2, and v3:

v1 = [-1, 4, 0, 0, 0, -1] v2 = [-5, 0, 2, -1, 0, -2] v3 = [5, 0, 0, 3, 3, 5]

The first step in the Gram-Schmidt procedure is to set u1 = v1, so:

u1 = v1 = [-1, 4, 0, 0, 0, -1]

Next, we calculate the projection of v2 onto u1, subtract it from v2, and set this as u2:

u2 = v2 - ((v2.u1)/(u1.u1)) * u1

We then calculate the projection of v3 onto the space spanned by u1 and u2, subtract it from v3, and set this as u3:

u3 = v3 - ((v3.u1)/(u1.u1)) * u1 - ((v3.u2)/(u2.u2)) * u2

The dot product <u1, v2> is the inner product of u1 and v2, which can be calculated as:

<u1, v2> = u1.v2

Similarly, <u1, u1> is the inner product of u1 and u1, which can be calculated as:

<u1, u1> = u1.u1

The dot product <u1, v3> is the inner product of u1 and v3, which can be calculated as:

<u1, v3> = u1.v3

The dot product <u2, v3> is the inner product of u2 and v3, which can be calculated as:

<u2, v3> = u2.v3

Finally, <u2, u2> is the inner product of u2 and u2, which can be calculated as:

<u2, u2> = u2.u2

To find the actual values of u2, u3, <u1, v2>, <u1, u1>, <u1, v3>, <u2, v3>, and <u2, u2>, you would need to perform these calculations.