[Math] 3 vectors in $\mathbb R^3$ which are linearly dependent, and two of them are linearly independent

vector-spaces

Find three vectors in $\mathbb R^3$ which are linearly dependent, and are such that any two of them are linearly independent?

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Take two non colinear vectors $u$ and $v$ and the third vector any linear combination of these vectors e.g $u+v$.

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[Math] How to show vectors are linearly independent

Any three vectors in a 2-dimensional space must be linearly dependent.

A common way to show that $n$ vectors in $\mathbb R^n$ are linearly independent is to make an $n$-by-$n$ matrix from the vectors and calculate the determinant. If the determinant is non-zero, the vectors are independent.

If the number of vectors is greater than the dimension of the vector space, the vectors must be linearly dependent. No calculation is needed. One way to find the dependency relationship is to place the vectors in a matrix augmented with the identity matrix. The reduced row echelon form will show the relationship in the augmented part, I believe.

If the number of vectors is less than the dimension of the vector space, you can make a matrix of the vectors, with each line being a row vector. Find the reduced row echelon form of the matrix. If any row becomes zero, the vectors are dependent. Otherwise, they are independent.

These are just common methods. Other methods also exist.

Finding out which vectors are linearly dependent

Just reduce it to row echelon form, keeping track of what you do. Start with $$ \begin{pmatrix} 1 & 2 & 3 & 1 &R_1\\ 2 & -1 & 1 & -3 &R_2\\ 1 & 3 & 4 & 1 &R_3\\ 3 & -1 & 2 & -2 &R_4 \end{pmatrix}$$

Then

\begin{pmatrix} 1 & 2 & 3 & 1 &R_1\\ 0 & -5 & -5 & -5 &R_2-2R_1\\ 0 & 1 & 1 & 0 &R_3 -R_1\\ 0 & -7 & -7 & -5 &R_4-3R_1 \end{pmatrix}

and continue in this fashion.

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Related Solutions

[Math] How to show vectors are linearly independent

Finding out which vectors are linearly dependent

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