Column vectors as entries of a column vector

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The representation in terms of vectors it is just a more compact way to express a linear system as the one you are given. $S$ in particular is the matrix of coefficients, which contains in its row and columns all the coefficients of the linear system, namely $s_{11},s_{12},...$ :

\begin{bmatrix} s_{11}\ s_{12}\ s_{13}\ ... \ s_{1n}\\ ........\\ ........\\ s_{n1}\ s_{n2} \ s_{n3} \ \dots s_{nn} \end{bmatrix}

in such a way that if you multiply the matrix $S$ times the vector $\bf{B}$ you obtain the RHS of your linear system.

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Updated on February 29, 2020

Comments

  • Admin
    Admin over 3 years

    I'm having a hard time understanding notation. Because of notation, I've lost hours and hours trying to understand a simple concept. I'm going to post a picture of the pdf that I've so that you can see exactly what I'm seeing and therefore have eventually the same doubts that I've (I hope not).

    enter image description here

    As far as I understand from the explanation $b_1, b_2, \cdots, b_n$ (in bold) of the basis $B$ are column vectors.

    I understand everything until the "or" (which comes after the system of equations). I don't understand this

    enter image description here

    I've a few questions that might help you to help me.

    1. What's the meaning of putting column vectors in a column vector. This quite blows my stupid mind.

    2. How is the linear system of equations

    enter image description here

    (before the "or") equivalent to this? Maybe a step my step explanation would clarify.

    Note that I understood well the linear system of equations and everything else, except, again, the equivalence between the linear system of equations and the equation that comes immediately after, which apparently should be equivalent.

  • la flaca
    la flaca over 7 years
    I see, you're right, is it possible that text is mistaken? It seems that they wrote $b_1, \ldots ,b_n$ instead of $v_1, \ldots ,v_n$ when they defined the basis $B$. Coordinates are usually represented with the letters $a,b,c$ and vectors with $v,w,u$, so i think they just put it the other way round at the begining of the text
  • Silvia
    Silvia over 7 years
    Again they tried to write it in a compact form. Just try to look at the first line of your linear system $ \bf{\tilde{b}}=s_{11} \bf{b_1} +\dots s_{1n} \bf{b_n}$ : this means that each of the entry of $ \bf{\tilde{b}}$ is a linear combination of the entries of the vectors of the other basis $\bf{b}$. If you want to write it explicitly each line of the system should be expanded to include the $\tilde{n}$ entries of each $ \bf{\tilde{b_1}}, \bf{\tilde{b_2}}, \bf{\tilde{b_n}}$. But you agree with me that it is quite tedious to write, that is why they work with vectors..
  • la flaca
    la flaca over 7 years
    Actually i don't think it has any meaning to put elements of $R^n$ as entries of a matrix (in this case of a column vector), entries of a matrix should be elements of a field
  • Silvia
    Silvia over 7 years
    So basically you don't have to multiply the S matrix times a' vector of vector' just consider the vector $\bf {b}$ as normal entrie of the a vector.
  • Silvia
    Silvia over 7 years
    So the formalism is the same, in a sense that whenever you a have a linear system e.g. $\tilde{x}= ax +by+cz, \tilde{y}= a'x +b'y+c'z, \tilde{z}= a''x +b''y+c''z$ you ca always write it down as $ \bf {\tilde{v}}= \bf A \bf v$ where ${\bf{\tilde{v}}}=(\tilde{x},\tilde{y}, \tilde{z})$ and ${\bf{v}}=(x,y,z)$ and A contains the coefficients a,b,c... In your case the only difference is that $x,y,z$, and $\tilde{x},\tilde{y}, \tilde{z}$ are vectors (it's a linear system of vectors instead of simple variables and the solution would be a vector in stead of a single value..). I hope this helps!
  • Silvia
    Silvia over 7 years
    Yes, it is correct. I didn't know which was the formula you were referring to in the other resource before. Anyway as a general advice try to multiply always the matrices and vector to see if you can recover the original linear system.