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MienTommy
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Homework Statement
\begin{pmatrix}
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0
\end{pmatrix}
Is this set a subspace of ℝ3
Homework Equations
The set must be closed under addition.
The set must be closed under multiplication.
The set must contain the zero vector.
The Attempt at a Solution
1. It obviously contains the zero vector (column 3)
2. \begin{pmatrix}
2\\
0\\
0\\
\end{pmatrix}
is a multiple of column 1.
\begin{pmatrix}
1 & 1 & 0 & 0 & 2\\
0 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 0 & 0
\end{pmatrix}
In order to check if this set contains the multiple of column 1, I set the 2 * column 1 equal to the matrix.
It row reduces to
\begin{pmatrix}
1 & 0 & 0 & 0 & 2\\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0
\end{pmatrix}
Wouldn't this be closed under multiplication? Since this contains a solution?
3. Closed under addition
Column 2 + column 3 becomes
\begin{pmatrix}
1\\
2\\
1\\
\end{pmatrix}It row reduces to
\begin{pmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 1\\
0 & 0 & 1 & 0 & 1
\end{pmatrix}
And wouldn't this be closed under addition? Since this contains a solution?
Edit: I have attached the problem and solution to the thread. I'm not sure how my teacher came across this solution.
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