Consider the following set of vectors. V₁ = v₂ = 3 + V3 = Let V₁, V₂, and v3 be (column) vectors in R3 and let A be the 3 x 3 matrix V₁ V₂ V3 with these vectors as its columns. Then V₁, V₂, and v3 are linearly dependent if and only if the homogeneous linear system with augmented matrix [A10] has a nontrivia solution. Consider the following equation. C₁ √ [1] +2₂2 3 -7 = 0 0 Solve for C₁, C₂, and C3. If a nontrivial solution exists, state it or state the general solution in terms of the parameter t. (If only the trivial solution exists, enter the trivial solution {C₁, C₂, C3} = {0, 0, 0}.) {C₁, C₂, C3} = < Determine if the vectors V₁, V₂, and v3 are linearly independent. The set of vectors is linearly dependent. The set of vectors is linearly independent.
Consider the following set of vectors. V₁ = v₂ = 3 + V3 = Let V₁, V₂, and v3 be (column) vectors in R3 and let A be the 3 x 3 matrix V₁ V₂ V3 with these vectors as its columns. Then V₁, V₂, and v3 are linearly dependent if and only if the homogeneous linear system with augmented matrix [A10] has a nontrivia solution. Consider the following equation. C₁ √ [1] +2₂2 3 -7 = 0 0 Solve for C₁, C₂, and C3. If a nontrivial solution exists, state it or state the general solution in terms of the parameter t. (If only the trivial solution exists, enter the trivial solution {C₁, C₂, C3} = {0, 0, 0}.) {C₁, C₂, C3} = < Determine if the vectors V₁, V₂, and v3 are linearly independent. The set of vectors is linearly dependent. The set of vectors is linearly independent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Vectors and Linear Dependence**
**Consider the following set of vectors:**
\[
\mathbf{v}_1 = \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 5 \\ 1 \\ 3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \\ -7 \\ 3 \end{bmatrix}
\]
Let \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) be column vectors in \(\mathbb{R}^3\) and let \(A\) be the \(3 \times 3\) matrix \(\begin{bmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{bmatrix}\) with these vectors as its columns. Then \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are linearly dependent if and only if the homogeneous linear system with augmented matrix \([A|0]\) has a nontrivial solution.
**Consider the following equation:**
\[
c_1 \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 5 \\ 1 \\ 3 \end{bmatrix} + c_3 \begin{bmatrix} 1 \\ -7 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]
Solve for \(c_1, c_2, c_3\). If a nontrivial solution exists, state the general solution in terms of the parameter \(t\). (If only the trivial solution exists, enter the trivial solution \(\{c_1, c_2, c_3\} = \{0, 0, 0\}\).)
\[
\{c_1, c_2, c_3\} = \{\underline{\phantom{0}}\}
\]
**Determine if the vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are linearly independent:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8fa7aa4-dafc-4c07-b274-462aa6ff800c%2Fd9c6ddc4-f696-4712-ac38-7aef771580b0%2Fk0fqdpp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vectors and Linear Dependence**
**Consider the following set of vectors:**
\[
\mathbf{v}_1 = \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 5 \\ 1 \\ 3 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \\ -7 \\ 3 \end{bmatrix}
\]
Let \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) be column vectors in \(\mathbb{R}^3\) and let \(A\) be the \(3 \times 3\) matrix \(\begin{bmatrix} \mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3 \end{bmatrix}\) with these vectors as its columns. Then \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are linearly dependent if and only if the homogeneous linear system with augmented matrix \([A|0]\) has a nontrivial solution.
**Consider the following equation:**
\[
c_1 \begin{bmatrix} 3 \\ 3 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 5 \\ 1 \\ 3 \end{bmatrix} + c_3 \begin{bmatrix} 1 \\ -7 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]
Solve for \(c_1, c_2, c_3\). If a nontrivial solution exists, state the general solution in terms of the parameter \(t\). (If only the trivial solution exists, enter the trivial solution \(\{c_1, c_2, c_3\} = \{0, 0, 0\}\).)
\[
\{c_1, c_2, c_3\} = \{\underline{\phantom{0}}\}
\]
**Determine if the vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are linearly independent:
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