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I know that brackets are used to group variables and numbers together. Do brackets serve the same purpose when they are used with other 'objects' such as matrices or vectors?

For example, when brackets are used with matrices does this expression, $A(B+C)$ still mean multiply matrix $A$ with the matrix sum $B+C$?

bob
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Kuskuba
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    Typically yes: parentheses are just there to indicate which operations to do first. There are circumstances where this isn't the case. For example, $(a, b)$ may refer to an ordered pair, or possibly an inner product. You also see $(a_n)_{n \in \Bbb{N}}$, or just $(a_n)$, to refer to a sequence. There are binomial coefficients, Pochhammer symbols, and a myriad of other exceptions. Context matters a whole heap. But generally, they are used to indicate which operations need to be done first. – Theo Bendit Oct 21 '23 at 02:41
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    (Round parentheses), [square brackets], {curly braces}. And yes, parentheses in expressions mean do the stuff inside the parentheses first. – coiso Oct 21 '23 at 02:41
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    $\langle$pointy braces or brackets$\rangle$? – Lutz Lehmann Oct 21 '23 at 06:59
  • ^Those are usually used for vectors to describe the direction from its tail to its tip. – Nate Oct 21 '23 at 23:13

2 Answers2

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As far as I know Yes.

The only times I have seen brackets behave differently depending on the field of math is for the specific examples below:

Geometry, where you can use them for points on a plane i.e. $(2, 1)$

Vectors, where you can display a vector as a position vector from the origin of a plane, allowing it to be expressed as a point as well i.e. $2i + 3j = (2,3)$ or it can be displayed as $2\choose3$

Complex Numbers, for yet again displaying a complex number as a point/position vector on the argand plane so e.g. $2 + 3i = (2,3)$

Matrices, when showing a matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, note that curly brackets still mean the same thing so $A(B+C) = AB + AC$

Combinatorics for choosing i.e. $3\choose2$ $= \frac{3!}{2!(3-2)!} = 3$

Functions, for passing the input i.e. $f(2)$ means input $x=2$ into the function $f(x)$ and not times the variable $f$ by $2$, note as well that this notation of putting the number in the bracket after the function also covers how $\mathrm{sin}(2) \neq 2\mathrm{sin}$ as instead of multiplication, the former means to pass the number $2$ to the sine function.

And set notation, so when creating a set you use curly brackets ${}$ so $N = \{2, 4, 6, 8, ...\}$ creates a set of all even numbers and not some infinite dimensional point

Joshua Finlayson
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  • Thank you for your detailed response! – Kuskuba Oct 22 '23 at 22:52
  • In the case where we have a function with matrices and vectors, such as a transformation, an inner product, the trace operator or the cross product etc. , applied to a set of matrices, would you view it in the same way as you do with for example polynomial function. For example for the function: Trace (A+B) , Is this read as the Trace of the sum of A+B, or the cross prodct , u x ( v + w ) - is this the same as the cross product between u and the sum of v and w? – Kuskuba Oct 22 '23 at 22:59
  • $u \times (v + w)$ is read as the cross product between u and the sum of v and w, it can also be factored out to $u \times v + u \times w$. The trace product is read like as the sum of $A + B$ and interestingly due to the function: Trace being linear, there is a proof for factoring it out into trace$(A) +$ trace$(B)$ – Joshua Finlayson Oct 24 '23 at 00:17
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Yes, absolutely! They always are meant to force a certain order of operation, i.e they always do what is in the brackets first.

Dheeraj Gujrathi
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J. Chapman
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  • Thank you! In the case where we have a function with matrices and vectors, such as a transformation, an inner product, the trace operator or the cross product etc. , applied to a set of matrices, would you view it in the same way as you do with for example polynomial function. For example for the function: Trace (A+B) , Is this read as the Trace of the sum of A+B, or the cross prodct , u x ( v + w ) - is this the same as the cross product between u and the sum of v and w? – Kuskuba Oct 22 '23 at 22:59