Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to study Linear Algebra :) In my textbook, the author said $$\begin{align} u_1=\begin{pmatrix}1 & 0 & 1 & 0 & -1\end{pmatrix} \end{align}$$ $$\begin{align} x_1=\begin{pmatrix}-1 \\ -2 \\ 1 \\ 0 \\ 0\end{pmatrix} \end{align}$$ $$\begin{align} u_1 \cdot x_1 = 0 \end{align}$$

However, I can't understand how I can do dot product with $u_1$ and $v_1$.

The form of $u_1$ is horizontally stretched, but $v_1$ is vertically stretched. Isn't it violating the definition of dot product of matrices?

share|improve this question
    
@Fant Aha, thanks :) –  Jonghwan Hyeon Dec 8 '12 at 12:43

2 Answers 2

up vote 2 down vote accepted

You are right, it's not the regular dot-product here, but the usual matrix multiplication.

The dot-product, also known as inner product, is a function $\mathcal{V}\times\mathcal{V}\longrightarrow\mathbb{R}$ Here you have $u_1\in\mathbb{R}^{1\times5}$ but $x_1\in\mathbb{R}^{5\times1}$, so the aren't really in the same vector space, which is why a inner product is a bit futile.

But the matrix multiplication $u_1\cdot x_1$ will give you the same value as the dot product of $u_1^T\cdot x_1$. The main difference here is that the dot product is commutative, thus $u_1^T\cdot x_1 = x_1\cdot u_1^T$, but the matrix product is not!

share|improve this answer

There can be defrined more kinds of 'products' among vectors and matrices.

The dot products for vectors is sometimes also dentoed by $\langle x,y\rangle$, and indeed it is defined primarily if $x$ and $y$ are coming from the 'same space', and usually the column vectors are preferred in usage. There is also a notion of transpose $M^T$ of a matrix $M$ which exchanges the indices and thus makes row vector from column vector and vice-versa.

Now, if $\bf x$ and $\bf y$ are both column vectors, with coordinates $(x_i)_i$ and $(y_i)_i$ then their dot product is $$\langle {\bf x},{\bf y}\rangle := x_1y_1+x_2y_2+x_3y_3+\ldots$$ And, if we have a matrix $A$ with rows $\pmatrix{{\bf a_1}\\{\bf a_2}\\{\bf a_3}\\ \vdots}$ and a matrix $B$ with columns $\pmatrix{{\bf b_1}&{\bf b_2}&{\bf b_3}&\ldots}$, then their matrix product is (can be) defined as the matrix $$A\cdot B:= (\langle {\bf a_i}^T,{\bf b_j}\rangle )_{i,j}\ .$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.