We had our first test and I'm trying to understand what it is I'm missing. I list each of the questions below, followed by what I answered.

 

1. Let v1=[ 1 ] and v2 = [ 1 ] --- Find a nonzero vector w that exists in R^3 such that {v1, v2, w} is linearly independent.

[ 1 ] [ 2 ]

[ 1 ] [ 3 ]

 

ans: w = [ 1 ] this was assuming that as long as the vector was a multiple of the of the vectors then the set would be linearly independent.

[ 4 ]

[ 6 ]

 

2. Find the general solution to the equation A*x = 0 (where x is a vector). Give your answer in parametric vector form.

 

A = [ 1 2 0 -2 0 ]

[ 0 0 1 2 0 ]

[ 0 0 0 0 1 ]

 

ans: this one I had no idea

Edited October 3, 201213 yr by D. Wellington

2. Find the general solution to the equation A*x = 0 (where x is a vector). Give your answer in parametric vector form.

 

A = [ 1 2 0 -2 0 ]

[ 0 0 1 2 0 ]

[ 0 0 0 0 1 ]

 

ans: this one I had no idea

 

[latex]\begin{pmatrix}1 & 2 & 0 & -2 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}a \\ b \\ c \\ d \\ e\end{pmatrix}=\begin{pmatrix}a+2b-2d \\ c+2d \\ e\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix}[/latex]

 

So, letting [latex]b=t,d=u[/latex], the general solution is

 

[latex]\mathbf{x}\,=\,\begin{pmatrix}-2t+2u \\ t \\ -2u \\ u \\ 0\end{pmatrix}[/latex]

We had our first test and I'm trying to understand what it is I'm missing. I list each of the questions below, followed by what I answered.

 

1. Let v1=[ 1 ] and v2 = [ 1 ] --- Find a nonzero vector w that exists in R^3 such that {v1, v2, w} is linearly independent.

[ 1 ] [ 2 ]

[ 1 ] [ 3 ]

 

ans: w = [ 1 ] this was assuming that as long as the vector was a multiple of the of the vectors then the set would be linearly independent.

[ 4 ]

[ 6 ]

I don't understand your vector representations. But from the definition of linear independence, it says the following, that the linear combination of the vectors; k_{1}v_{1} + k_{2}v_{2} + k_{3}w = 0, if k_{1}=k_{2}=k_{3}=0 solving this equation will give your solution.

Edited May 17, 201312 yr by shah_nosrat