1. ## Basis Orthogonal complement

I need to find a basis for the orthogonal complement of the subspace $\begin{array}{cc}t\\-t\\3t\end{array}$ that is orthogonal to the basis vector obtained by letting t = 1.

Ok so i think i need to find a vector that is orthogonal to this one, which means that i just dot product with another vector. I just don't know what vector to use, or exactly what to do with the t bit.... Any help will be appreciated thank you in advance.

2. Originally Posted by Orbent
I need to find a basis for the orthogonal complement of the subspace $\begin{array}{cc}t\\-t\\3t\end{array}$ that is orthogonal to the basis vector obtained by letting t = 1.

Ok so i think i need to find a vector that is orthogonal to this one, which means that i just dot product with another vector. I just don't know what vector to use, or exactly what to do with the t bit.... Any help will be appreciated thank you in advance.

I'm almost 100% sure I know what you meant, but college students must learn how to properly ask questions: what vector space are you talking about? What inner (or "dot") product is defined there? Why didn't you write your vector as $\begin{pmatrix}t\\\!\!\!-t\\3t\end{pmatrix}$ and not like that weird-looking thing?...

Tonio

3. Well i didn't know how to put $\begin{pmatrix}t\\\!\!\!-t\\3t\end{pmatrix}$ in. I am in engineering i learned about "dot product" before inner product so i associate Euclidean inner product with dot product. Since no vector space is mentioned in the question it is probably ok to assume it's euclidean. That being said i don't really know much about linear algebra, it is by far my weakest course. I hope i cleared everything up for you.

4. Okay, "letting t= 1" in that gives $\begin{pmatrix}1 \\ -1\\ 3\end{pmatrix}$. A vector, $\begin{pmatrix}a \\ b\\ c\end{pmatrix}$, will be orthogonal to that if and only if $\begin{pmatrix}a \\ b\\ c\end{pmatrix}\cdot\begin{pmatrix}1 \\ -1 \\ 3\end{pmatrix}= a- b+ 3c= 0$

From a- b+ 3c= 0, b= a+ 3c so $\begin{pmatrix}a \\ b\\ c\end{pmatrix}= \begin{pmatrix}a \\ a+ 3c \\ c\end{pmatrix}= a\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}+ c\begin{pmatrix}0 \\ 3 \\ 1\end{pmatrix}$.

5. Thank you halls, i was heading along the right track, just couldn't think straight. Thank very much!