I dont understand vector matrices please help

Livingstone

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Nov 24, 2018
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so the question goes

Let [FONT=MathJax_Math]V[/FONT] be the vector space of symmetric [FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Main]2[/FONT] matrices and [FONT=MathJax_Math]W[/FONT] be the subspace
[FONT=MathJax_Main]

W={[-5 0][4 0]}
||||||||||[0 -3][0 2]||||

[/FONT]​
then they say find an element Y in V that is not in W?

please dont solve it for me and please don't let the mods come down on me for asking a question. could someone just give me an explanation of what they're wanting me to do
as well as a hint of how to do it.
 
Last edited:
The set, W, as you show it is not a subspace. You must mean the subspace spanned by the matrices \(\displaystyle \begin{bmatrix}-5 & 0 \\ 0 & -3 \end{bmatrix}\) and \(\displaystyle \begin{bmatrix}4 & 0 \\ 0 & 2 \end{bmatrix}\).

That subspace consists of all matrices of the form \(\displaystyle a\begin{bmatrix}-5 & 0 \\ 0 & -3 \end{bmatrix}+ b\begin{bmatrix}4 & 0 \\ 0 & 2 \end{bmatrix}= \begin{bmatrix}-5a+ 4b & 0 \\ 0 & -3a+ 2b \end{bmatrix}\).

Notice that those have both non-main-diagonal elements 0. Your space, V, is "all symmetric 2 by 2 matrices" so the "non-main-diagonal" elements must be the same but not necessarily 0. "An element, Y, that is in V but not in W" is a symmetric 2 by 2 matrix where the "non-main-diagonal" elements are not 0.
 
How would you determine that?
Do you really understand that
That subspace consists of all matrices of the form \(\displaystyle a\begin{bmatrix}-5 & 0 \\ 0 & -3 \end{bmatrix}+ b\begin{bmatrix}4 & 0 \\ 0 & 2 \end{bmatrix}= \begin{bmatrix}-5a+ 4b & 0 \\ 0 & -3a+ 2b \end{bmatrix}~?\)

If you do then find a \(\displaystyle 2\times 2\) symmetric matrix that does no have that form.
 
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