Determine whether or not the following vectors in R3 are linearly dependent: u = (1,2,5), v = (2,5,1), and w…

Question

Determine whether or not the following vectors in R3 are linearly dependent: u = (1,2,5), v = (2,5,1), and w = (1,5,2).

ScanSolve AI Tutor · Accepted Answer

let's determine the linear dependence of these vectors.

3. Determine whether or not the following vectors in $\mathbb{R}^3$ are linearly dependent: $u = (1,2,5)$ , $v = (2,5,1)$ , and $w = (1,5,2)$ .

To determine if the vectors are linearly dependent, we can form a matrix with these vectors as columns and calculate its determinant. If the determinant is zero, the vectors are linearly dependent. If it's non-zero, they are linearly independent.

Form the matrix $A$ with the given vectors as columns.
$A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 5 & 5 \\ 5 & 1 & 2 \end{bmatrix}$
Calculate the determinant of matrix $A$ .
We will use cofactor expansion along the first row:
$\det(A) = 1 \cdot \det \begin{bmatrix} 5 & 5 \\ 1 & 2 \end{bmatrix} - 2 \cdot \det \begin{bmatrix} 2 & 5 \\ 5 & 2 \end{bmatrix} + 1 \cdot \det \begin{bmatrix} 2 & 5 \\ 5 & 1 \end{bmatrix}$
Compute the $2 \times 2$ determinants.
$\det \begin{bmatrix} 5 & 5 \\ 1 & 2 \end{bmatrix} = (5 \cdot 2) - (5 \cdot 1) = 10 - 5 = 5$ $\det \begin{bmatrix} 2 & 5 \\ 5 & 2 \end{bmatrix} = (2 \cdot 2) - (5 \cdot 5) = 4 - 25 = -21$ $\det \begin{bmatrix} 2 & 5 \\ 5 & 1 \end{bmatrix} = (2 \cdot 1) - (5 \cdot 5) = 2 - 25 = -23$
Substitute these values back into the determinant formula and simplify.
$\det(A) = 1 \cdot (5) - 2 \cdot (-21) + 1 \cdot (-23)$ $\det(A) = 5 + 42 - 23$ $\det(A) = 47 - 23$ $\det(A) = 24$
Conclude based on the determinant value.
Since $\det(A) = 24 \neq 0$ , the vectors are linearly independent.
$\boxed{\text{The vectors are linearly independent}}$

Determine whether or not the following vectors in R3 are linearly dependent: u = (1,2,5), v = (2,5,1), and w = (1,5,2).

Step-by-step solution

Form the matrix $A$ with the given vectors as columns.

Calculate the determinant of matrix $A$ .

Compute the $2 \times 2$ determinants.

Substitute these values back into the determinant formula and simplify.

Conclude based on the determinant value.

Still stuck on this one?

More Mathematics Questions

Determine whether or not the following vectors in R3 are linearly dependent: u = (1,2,5), v = (2,5,1), and w = (1,5,2).

Step-by-step solution

Form the matrix AAA with the given vectors as columns.

Calculate the determinant of matrix AAA.

Compute the 2×22 \times 22×2 determinants.

Substitute these values back into the determinant formula and simplify.

Conclude based on the determinant value.

Still stuck on this one?

More Mathematics Questions

Form the matrix $A$ with the given vectors as columns.

Calculate the determinant of matrix $A$ .

Compute the $2 \times 2$ determinants.