Question

(after 2.2) find all values z1 and z2 such that (2, −1, 3), (1, 2, 2), and (−4, z1, z2) do not span r 3 .

Answer 1

To ensure the vectors do not span R^3, they must be linearly dependent. This occurs if the determinant of the matrix formed by the vectors is zero, leading to the equation 8*z2 − 15*z1 − 8 = 0, which provides the relationship between z1 and z2 for the vectors to be linearly dependent.

Step-by-step explanation:

To find all values z1 and z2 such that the vectors (2, −1, 3), (1, 2, 2), and (−4, z1, z2) do not span R^3, we need to determine when these vectors are linearly dependent. Three vectors in R^3 are linearly dependent if there exists a non-trivial solution to the equation c1*v1 + c2*v2 + c3*v3 = 0, where v1, v2, and v3 are the vectors and c1, c2, and c3 are constants not all zero.

Let's set up the matrix whose rows are our vectors:

1. (2, −1, 3)
2. (1, 2, 2)
3. (−4, z1, z2)

For the vectors to be linearly dependent, the determinant of this matrix must be zero.

The determinant can be calculated as:

Det = (2*(2*z2 − 2*z1) − (1*(z1*3 − −4*z2) + (−4*(4 + 2*z1))

For the vectors not to span R^3, this determinant must be equal to zero. This gives us an equation in terms of z1 and z2:

4*z2 − 4*z1 − 3*z1 + 4*z2 − 8 − 8*z1 = 0

Simplifying this gives:

8*z2 − 15*z1 − 8 = 0

This equation gives us the relationship between z1 and z2. Any pair (z1, z2) that satisfies this equation means the vectors do not span R^3.

Answer 2

To find values of z1 and z2 such that the given vectors do not span R3, we can set up an augmented matrix and row-reduce it to check for non-trivial solutions. If there are no non-trivial solutions, it means that any values of z1 and z2 will span R3.

Step-by-step explanation:

To determine the values of z1 and z2 such that the vectors (2, −1, 3), (1, 2, 2), and (−4, z1, z2) do not span R3, we need to find a non-trivial linear combination of these vectors that equals zero.

By setting up an augmented matrix and performing row operations to row-reduce the matrix, we can determine if there are any non-trivial solutions. If there are non-trivial solutions, it means that there exist values of z1 and z2 that do not span R3.

Let's set up the augmented matrix:

[2 1 -4 | 0]

Row-reducing the matrix:

[1 1/2 -2 | 0]

We can see that there are no free variables, which means that the system has only the trivial solution. Therefore, for any values of z1 and z2, the vectors (2, −1, 3), (1, 2, 2), and (−4, z1, z2) span the entire R3.