Let u=(2,0,k,-1) v=(-4,0,-3,2) and w= (0,1,0,0). Each answer must be justified(there is no answer where k=nothing)

a) find all values of k, (if any) for which u is orthogonal to v

b) Find all values of k (if any) for which the set {u,v,w} is linearly independent.

Find all values of k (if any) for which the set {u,v,w} is a basis of R4

a) To determine if u is orthogonal to v, we need the dot product of u and v to be zero.

The dot product of u and v is calculated as follows:

u · v = (2)(-4) + (0)(0) + (k)(-3) + (-1)(2) = -8 - 2k

For u to be orthogonal to v, the dot product u · v must equal zero:

-8 - 2k = 0

To solve the equation, we will isolate k:

-2k = 8

k = -4

Therefore, the value of k for which u is orthogonal to v is k = -4.

b) To determine if the set {u, v, w} is linearly independent, we need to check if there exist values of k such that the set can only be represented by the trivial solution.

We can create a matrix using the vectors u, v, and w as its columns:

| 2 -4 0 |
| 0 0 1 |
| k -3 0 |
| -1 2 0 |

To find the values of k for which the set {u, v, w} is linearly independent, we can perform row reduction on the matrix to obtain its row-echelon form:

| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
| 0 0 0 |

If the matrix reduces to the identity matrix, then the set is linearly independent. If the rows reduce to all zeros except for the last row, then the set is linearly dependent.

We can solve this system of equations to determine the values of k that result in the matrix reducing to the identity matrix:

Row 1: 2(1) - 4(0) + 0(0) = 1 --> 2 = 1 (contradiction)
Row 2: 0(1) - 0(0) + 1(0) = 1 --> 0 = 1 (contradiction)

Since the equations for Row 1 and Row 2 lead to contradictions, there are no values of k for which the set {u, v, w} is linearly independent.

Therefore, for any value of k, the set {u, v, w} is linearly dependent.

c) To determine if the set {u, v, w} is a basis of R^4, we need to check if the set is linearly independent and spans R^4.

Since we have already determined that the set {u, v, w} is linearly dependent for any value of k, it cannot be a basis of R^4.

Therefore, there are no values of k for which the set {u, v, w} is a basis of R^4.