Introduction to Linear Algebra Calculator

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Example

Created on 2024-06-20Asked by Chloe Hill (Solvelet student)

Determine if the following vectors are linearly independent:

\mathbf{v_1} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \mathbf{v_2} = \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix}, \mathbf{v_3} = \begin{pmatrix} 7 \\ 8 \\ 9 \end{pmatrix}

Solution

To determine if the vectors

\mathbf{v_1} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}

\mathbf{v_2} = \begin{pmatrix} 4 \ 5 \ 6 \end{pmatrix}

, and

\mathbf{v_3} = \begin{pmatrix} 7 \ 8 \ 9 \end{pmatrix}

are linearly independent, we form a matrix with these vectors as columns and find its determinant:

A = \begin{pmatrix} 1 & 4 & 7 \ 2 & 5 & 8 \ 3 & 6 & 9 \end{pmatrix}

Calculate the determinant of matrix

A

\text{det}(A) = \begin{vmatrix} 1 & 4 & 7 \ 2 & 5 & 8 \ 3 & 6 & 9 \end{vmatrix}

Using cofactor expansion along the first row:

\text{det}(A) = 1 \cdot \begin{vmatrix} 5 & 8 \ 6 & 9 \end{vmatrix} - 4 \cdot \begin{vmatrix} 2 & 8 \ 3 & 9 \end{vmatrix} + 7 \cdot \begin{vmatrix} 2 & 5 \ 3 & 6 \end{vmatrix}

Calculate the 2x2 determinants:

\begin{vmatrix} 5 & 8 \ 6 & 9 \end{vmatrix} = 5 \cdot 9 - 8 \cdot 6 = 45 - 48 = -3

\begin{vmatrix} 2 & 8 \ 3 & 9 \end{vmatrix} = 2 \cdot 9 - 8 \cdot 3 = 18 - 24 = -6

\begin{vmatrix} 2 & 5 \ 3 & 6 \end{vmatrix} = 2 \cdot 6 - 5 \cdot 3 = 12 - 15 = -3

Substitute these into the determinant calculation:

\text{det}(A) = 1(-3) - 4(-6) + 7(-3) = -3 + 24 - 21 = 0

Since the determinant of

A

is zero, the vectors are linearly dependent. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Alexander Nelson on Solvelet

1. Determine whether the vectors

\{\langle 1, 2 \rangle, \langle 3, 4 \rangle, \langle 5, 6 \rangle\}

form a linearly independent set,2. Find the original time-domain signal

f(t)

given its frequency-domain representation

F(\omega) = e^{-\omega}

DefinitionYou have some enjoyable work to study vector spaces, linear transformations, and systems of linear equations in the introduction to linear algebra. Examples among them are Vectors, Matrices, Determinants, Eigenvalues and Eigenvectors. Linear algebra is a branch of mathematics that is widely used throughout the natural and social sciences,intended to solve problems dealing with linear systems and transformations. E.g. solving a linear equation system Ax=b through matrix methods you have x=A−1b if A is invertible

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